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Page No 299:
Question 1:
sin 2x
ANSWER:
The anti derivative of sin 2x is a function of x whose derivative is sin 2x.
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7381/NCERT_Solution_Math_Chapter_7_final_html_m697e7f3e.gif)
Therefore, the anti derivative of
Page No 299:
Question 2:
Cos 3x
ANSWER:
The anti derivative of cos 3x is a function of x whose derivative is cos 3x.
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7383/NCERT_Solution_Math_Chapter_7_final_html_37502309.gif)
Therefore, the anti derivative of .
Page No 299:
Question 3:
e2x
ANSWER:
The anti derivative of e2x is the function of x whose derivative is e2x.
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7385/NCERT_Solution_Math_Chapter_7_final_html_m28a7365e.gif)
Therefore, the anti derivative of .
Page No 299:
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7386/NCERT_Solution_Math_Chapter_7_final_html_d228bea.gif)
ANSWER:
The anti derivative of is the function of x whose derivative is
.
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7386/NCERT_Solution_Math_Chapter_7_final_html_m22680595.gif)
Therefore, the anti derivative of .
Page No 299:
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7387/NCERT_Solution_Math_Chapter_7_final_html_m64ac22d.gif)
ANSWER:
The anti derivative of is the function of x whose derivative is
.
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7387/NCERT_Solution_Math_Chapter_7_final_html_37d39277.gif)
Therefore, the anti derivative of is
.
Page No 299:
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7388/NCERT_Solution_Math_Chapter_7_final_html_m1fde19c0.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7388/NCERT_Solution_Math_Chapter_7_final_html_m1fde19c0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7388/NCERT_Solution_Math_Chapter_7_final_html_55f021fe.gif)
Page No 299:
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7389/NCERT_Solution_Math_Chapter_7_final_html_76cc6ec4.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7389/NCERT_Solution_Math_Chapter_7_final_html_76cc6ec4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7389/NCERT_Solution_Math_Chapter_7_final_html_5e8acc03.gif)
Page No 299:
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7390/NCERT_Solution_Math_Chapter_7_final_html_5ebb449b.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7390/NCERT_Solution_Math_Chapter_7_final_html_5ebb449b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7390/NCERT_Solution_Math_Chapter_7_final_html_146e5d64.gif)
Page No 299:
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7391/NCERT_Solution_Math_Chapter_7_final_html_4b2d046b.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7391/NCERT_Solution_Math_Chapter_7_final_html_4b2d046b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7391/NCERT_Solution_Math_Chapter_7_final_html_1319c4b3.gif)
Page No 299:
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7392/NCERT_Solution_Math_Chapter_7_final_html_m21b097a0.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7392/NCERT_Solution_Math_Chapter_7_final_html_m21b097a0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7392/NCERT_Solution_Math_Chapter_7_final_html_353d003.gif)
Page No 299:
Question 11:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7393/NCERT_Solution_Math_Chapter_7_final_html_m45687b1f.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7393/NCERT_Solution_Math_Chapter_7_final_html_m45687b1f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7393/NCERT_Solution_Math_Chapter_7_final_html_m60c06e96.gif)
Page No 299:
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7394/NCERT_Solution_Math_Chapter_7_final_html_m352a7734.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7394/NCERT_Solution_Math_Chapter_7_final_html_m352a7734.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7394/NCERT_Solution_Math_Chapter_7_final_html_75ad4a9d.gif)
Page No 299:
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7395/NCERT_Solution_Math_Chapter_7_final_html_m762c7093.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7395/NCERT_Solution_Math_Chapter_7_final_html_m762c7093.gif)
On dividing, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7395/NCERT_Solution_Math_Chapter_7_final_html_19f077a8.gif)
Page No 299:
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7396/NCERT_Solution_Math_Chapter_7_final_html_1b1677ba.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7396/NCERT_Solution_Math_Chapter_7_final_html_1b1677ba.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7396/NCERT_Solution_Math_Chapter_7_final_html_982111b.gif)
Page No 299:
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7397/NCERT_Solution_Math_Chapter_7_final_html_ef118c8.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7397/NCERT_Solution_Math_Chapter_7_final_html_ef118c8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7397/NCERT_Solution_Math_Chapter_7_final_html_2b7fadb0.gif)
Page No 299:
Question 16:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7398/NCERT_Solution_Math_Chapter_7_final_html_70eedab9.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7398/NCERT_Solution_Math_Chapter_7_final_html_70eedab9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7398/NCERT_Solution_Math_Chapter_7_final_html_m155aa073.gif)
Page No 299:
Question 17:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7399/NCERT_Solution_Math_Chapter_7_final_html_m3513f78e.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7399/NCERT_Solution_Math_Chapter_7_final_html_m3513f78e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7399/NCERT_Solution_Math_Chapter_7_final_html_552b2cfe.gif)
Page No 299:
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7400/NCERT_Solution_Math_Chapter_7_final_html_55f8b69e.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7400/NCERT_Solution_Math_Chapter_7_final_html_55f8b69e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7400/NCERT_Solution_Math_Chapter_7_final_html_m6170702b.gif)
Page No 299:
Question 19:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7401/NCERT_Solution_Math_Chapter_7_final_html_69cfa7d9.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7401/NCERT_Solution_Math_Chapter_7_final_html_m5d1cbb50.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7401/NCERT_Solution_Math_Chapter_7_final_html_m8779d63.gif)
Page No 299:
Question 20:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7402/NCERT_Solution_Math_Chapter_7_final_html_m4223465.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7402/NCERT_Solution_Math_Chapter_7_final_html_m4223465.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7402/NCERT_Solution_Math_Chapter_7_final_html_m63d2802c.gif)
Page No 299:
Question 21:
The anti derivative of equals
![](https://ddtarget.com/wp-content/uploads/2024/01/image-59.png)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7403/NCERT_Solution_Math_Chapter_7_final_html_m53d4ecad.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7403/NCERT_Solution_Math_Chapter_7_final_html_7839ddd7.gif)
Hence, the correct answer is C.
Page No 299:
Question 22:
If such that f(2) = 0, then f(x) is
(A) (B)
(C) (D)
ANSWER:
It is given that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7404/NCERT_Solution_Math_Chapter_7_final_html_3ad53d45.gif)
∴Anti derivative of
∴
Also,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7404/NCERT_Solution_Math_Chapter_7_final_html_m601912de.gif)
Hence, the correct answer is A.
Page No 304:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7405/NCERT_Solution_Math_Chapter_7_final_html_2daa2dc2.gif)
ANSWER:
Let = t
∴2x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7405/NCERT_Solution_Math_Chapter_7_final_html_285bfdcb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7405/NCERT_Solution_Math_Chapter_7_final_html_4f3f798c.gif)
Page No 304:
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7406/NCERT_Solution_Math_Chapter_7_final_html_4fda87ef.gif)
ANSWER:
Let log |x| = t
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7406/NCERT_Solution_Math_Chapter_7_final_html_52ef2a81.gif)
Page No 304:
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_m61551e94.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_m4370fbad.gif)
Let 1 + log x = t
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_m2ebe9270.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7407/NCERT_Solution_Math_Chapter_7_final_html_1cd43b92.gif)
Page No 304:
Question 4:
sin x ⋅ sin (cos x)
ANSWER:
sin x ⋅ sin (cos x)
Let cos x = t
∴ −sin x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7408/NCERT_Solution_Math_Chapter_7_final_html_69fb365b.gif)
Page No 304:
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7409/NCERT_Solution_Math_Chapter_7_final_html_42180c1b.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7409/NCERT_Solution_Math_Chapter_7_final_html_1400fe22.gif)
Let
∴ 2adx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7409/NCERT_Solution_Math_Chapter_7_final_html_396f0375.gif)
Page No 304:
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7411/NCERT_Solution_Math_Chapter_7_final_html_m6a9188e.gif)
ANSWER:
Let ax + b = t
⇒ adx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7411/NCERT_Solution_Math_Chapter_7_final_html_m5727662a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7411/NCERT_Solution_Math_Chapter_7_final_html_m10c63328.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7411/NCERT_Solution_Math_Chapter_7_final_html_m31dc4753.gif)
Page No 304:
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7414/NCERT_Solution_Math_Chapter_7_final_html_6cb9b43.gif)
ANSWER:
Let
∴ dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7414/NCERT_Solution_Math_Chapter_7_final_html_4c24a780.gif)
Page No 304:
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7416/NCERT_Solution_Math_Chapter_7_final_html_4e827515.gif)
ANSWER:
Let 1 + 2x2 = t
∴ 4xdx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7416/NCERT_Solution_Math_Chapter_7_final_html_m34de4fbb.gif)
Page No 304:
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7417/NCERT_Solution_Math_Chapter_7_final_html_m4b12630c.gif)
ANSWER:
Let
∴ (2x + 1)dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7417/NCERT_Solution_Math_Chapter_7_final_html_76f6bdb0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7417/NCERT_Solution_Math_Chapter_7_final_html_m75db78a0.gif)
Page No 304:
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_520c9935.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_4563a36f.gif)
Let
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_2d5d2240.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7420/NCERT_Solution_Math_Chapter_7_final_html_1a4ba108.gif)
Page No 304:
Question 11:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7422/NCERT_Solution_Math_Chapter_7_final_html_44d0f94f.gif)
ANSWER:
![](https://ddtarget.com/wp-content/uploads/2024/01/image-61.png)
Page No 304:
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7424/NCERT_Solution_Math_Chapter_7_final_html_m132ed99c.gif)
ANSWER:
Let
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7424/NCERT_Solution_Math_Chapter_7_final_html_m452c22f6.gif)
Page No 304:
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7427/NCERT_Solution_Math_Chapter_7_final_html_3d95f002.gif)
ANSWER:
Let
∴ 9x2dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7427/NCERT_Solution_Math_Chapter_7_final_html_m649abaa8.gif)
Page No 304:
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7428/NCERT_Solution_Math_Chapter_7_final_html_78cb1997.gif)
ANSWER:
Let log x = t
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7428/NCERT_Solution_Math_Chapter_7_final_html_3e19e712.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7428/NCERT_Solution_Math_Chapter_7_final_html_a99ad2.gif)
Page No 304:
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7442/NCERT_Solution_Math_Chapter_7_final_html_m42ca294.gif)
ANSWER:
Let
∴ −8x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7442/NCERT_Solution_Math_Chapter_7_final_html_m15aecbfc.gif)
Page No 304:
Question 16:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7444/NCERT_Solution_Math_Chapter_7_final_html_2939b26a.gif)
ANSWER:
Let
∴ 2dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7444/NCERT_Solution_Math_Chapter_7_final_html_m513791a7.gif)
Page No 304:
Question 17:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7447/NCERT_Solution_Math_Chapter_7_final_html_6ddfda2d.gif)
ANSWER:
Let
∴ 2xdx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7447/NCERT_Solution_Math_Chapter_7_final_html_m650f6bc9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7447/NCERT_Solution_Math_Chapter_7_final_html_7c9ba91.gif)
Page No 305:
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_m481a6bf9.gif)
ANSWER:
Let
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_277fdce0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7449/NCERT_Solution_Math_Chapter_7_final_html_m666c0c5.gif)
Page No 305:
Question 19:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_4bdb887f.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_4bdb887f.gif)
Dividing numerator and denominator by ex, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_m6cb48a41.gif)
Let
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_m449ce1c0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7452/NCERT_Solution_Math_Chapter_7_final_html_79d625cd.gif)
Page No 305:
Question 20:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m8dd01a5.gif)
ANSWER:
Let
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m3314c039.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7453/NCERT_Solution_Math_Chapter_7_final_html_m5a357383.gif)
Page No 305:
Question 21:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7456/NCERT_Solution_Math_Chapter_7_final_html_60990151.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7456/NCERT_Solution_Math_Chapter_7_final_html_60990151.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7456/NCERT_Solution_Math_Chapter_7_final_html_50852ccc.gif)
Let 2x − 3 = t
∴ 2dx = dt
![](https://ddtarget.com/wp-content/uploads/2024/01/image-60.png)
Page No 305:
Question 22:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7458/NCERT_Solution_Math_Chapter_7_final_html_30c46dae.gif)
ANSWER:
Let 7 − 4x = t
∴ −4dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7458/NCERT_Solution_Math_Chapter_7_final_html_5dbac121.gif)
Page No 305:
Question 23:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_59d50245.gif)
ANSWER:
Let
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_d934bc5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7460/NCERT_Solution_Math_Chapter_7_final_html_m615ee39c.gif)
Page No 305:
Question 24:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_m1e861f.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_m1b300ce0.gif)
Let
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7466/NCERT_Solution_Math_Chapter_7_final_html_29ad5f75.gif)
Page No 305:
Question 25:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_m67c449bc.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_6fe644c1.gif)
Let
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7471/NCERT_Solution_Math_Chapter_7_final_html_56990d0f.gif)
Page No 305:
Question 26:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7477/NCERT_Solution_Math_Chapter_7_final_html_m3f7ab82.gif)
ANSWER:
Let
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7477/NCERT_Solution_Math_Chapter_7_final_html_1195f5d9.gif)
Page No 305:
Question 27:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7480/NCERT_Solution_Math_Chapter_7_final_html_396850ae.gif)
ANSWER:
Let sin 2x = t
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7480/NCERT_Solution_Math_Chapter_7_final_html_103e3be3.gif)
Page No 305:
Question 28:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7483/NCERT_Solution_Math_Chapter_7_final_html_m24b03964.gif)
ANSWER:
Let
∴ cos x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7483/NCERT_Solution_Math_Chapter_7_final_html_4631f4db.gif)
Page No 305:
Question 29:
cot x log sin x
ANSWER:
Let log sin x = t
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7485/NCERT_Solution_Math_Chapter_7_final_html_m3e62cda.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7485/NCERT_Solution_Math_Chapter_7_final_html_8ef846c.gif)
Page No 305:
Question 30:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7487/NCERT_Solution_Math_Chapter_7_final_html_5910cc29.gif)
ANSWER:
Let 1 + cos x = t
∴ −sin x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7487/NCERT_Solution_Math_Chapter_7_final_html_537be00f.gif)
Page No 305:
Question 31:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7489/NCERT_Solution_Math_Chapter_7_final_html_594e5e18.gif)
ANSWER:
Let 1 + cos x = t
∴ −sin x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7489/NCERT_Solution_Math_Chapter_7_final_html_m24732181.gif)
Page No 305:
Question 32:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7490/NCERT_Solution_Math_Chapter_7_final_html_m70797b4a.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7490/NCERT_Solution_Math_Chapter_7_final_html_42b7f736.gif)
Let sin x + cos x = t ⇒ (cos x − sin x) dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7490/NCERT_Solution_Math_Chapter_7_final_html_e9f2ce0.gif)
Page No 305:
Question 33:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7492/NCERT_Solution_Math_Chapter_7_final_html_70acf21f.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7492/NCERT_Solution_Math_Chapter_7_final_html_m1fecaf67.gif)
Put cos x − sin x = t ⇒ (−sin x − cos x) dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7492/NCERT_Solution_Math_Chapter_7_final_html_1c1841d8.gif)
Page No 305:
Question 34:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7494/NCERT_Solution_Math_Chapter_7_final_html_m3cd2a5e1.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7494/NCERT_Solution_Math_Chapter_7_final_html_m441fc4b5.gif)
Page No 305:
Question 35:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7497/NCERT_Solution_Math_Chapter_7_final_html_6ef9e05f.gif)
ANSWER:
Let 1 + log x = t
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7497/NCERT_Solution_Math_Chapter_7_final_html_m79272764.gif)
Page No 305:
Question 36:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_663e2a5d.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_m7e821b61.gif)
Let
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7499/NCERT_Solution_Math_Chapter_7_final_html_m2f765048.gif)
Page No 305:
Question 37:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_3d29e0a5.gif)
ANSWER:
Let x4 = t
∴ 4x3 dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_m4c5a1b01.gif)
Let
∴
From (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_626f7efb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7501/NCERT_Solution_Math_Chapter_7_final_html_m776b9de8.gif)
Page No 305:
Question 38:
equals
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_21be7c9a.gif)
ANSWER:
Let
∴
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_m3b428ede.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7504/NCERT_Solution_Math_Chapter_7_final_html_3b394c63.gif)
Hence, the correct answer is D.
Page No 305:
Question 39:
equals
A.
B.
C.
![](https://ddtarget.com/wp-content/uploads/2024/01/image-64.png)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7507/NCERT_Solution_Math_Chapter_7_final_html_m3109d37e.gif)
Hence, the correct answer is B.
Page No 307:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7510/NCERT_Solution_Math_Chapter_7_final_html_158e4d81.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7510/NCERT_Solution_Math_Chapter_7_final_html_m25ad4f6f.gif)
Page No 307:
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7520/NCERT_Solution_Math_Chapter_7_final_html_m79d80b67.gif)
ANSWER:
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7520/NCERT_Solution_Math_Chapter_7_final_html_4e6849bf.gif)
Page No 307:
Question 3:
cos 2x cos 4x cos 6x
ANSWER:
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7523/NCERT_Solution_Math_Chapter_7_final_html_m453c4158.gif)
Page No 307:
Question 4:
sin3 (2x + 1)
ANSWER:
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7525/NCERT_Solution_Math_Chapter_7_final_html_2257b9f2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7525/NCERT_Solution_Math_Chapter_7_final_html_m5795a85f.gif)
Page No 307:
Question 5:
sin3x cos3x
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7527/NCERT_Solution_Math_Chapter_7_final_html_m73e75233.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7527/NCERT_Solution_Math_Chapter_7_final_html_69753c1b.gif)
Page No 307:
Question 6:
sin x sin 2x sin 3x
ANSWER:
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7529/NCERT_Solution_Math_Chapter_7_final_html_4b01ff05.gif)
Page No 307:
Question 7:
sin 4x sin 8x
ANSWER:
![](https://ddtarget.com/wp-content/uploads/2024/01/image-63.png)
Page No 307:
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7533/NCERT_Solution_Math_Chapter_7_final_html_4d6e72c3.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/Kayako_FKY%2019079408.gif)
Page No 307:
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7535/NCERT_Solution_Math_Chapter_7_final_html_7705fbfd.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7535/NCERT_Solution_Math_Chapter_7_final_html_m316e357f.gif)
Page No 307:
Question 10:
sin4x
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7536/NCERT_Solution_Math_Chapter_7_final_html_m48648a1c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7536/NCERT_Solution_Math_Chapter_7_final_html_m31f5f804.gif)
Page No 307:
Question 11:
cos4 2x
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7538/NCERT_Solution_Math_Chapter_7_final_html_m2f3cf2d6.gif)
Page No 307:
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7540/NCERT_Solution_Math_Chapter_7_final_html_639207d4.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7540/NCERT_Solution_Math_Chapter_7_final_html_1b7154f9.gif)
Page No 307:
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7543/NCERT_Solution_Math_Chapter_7_final_html_m3bf74025.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7543/NCERT_Solution_Math_Chapter_7_final_html_m3df62037.gif)
Page No 307:
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7544/NCERT_Solution_Math_Chapter_7_final_html_m2f3df7c7.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7544/NCERT_Solution_Math_Chapter_7_final_html_m3ed05535.gif)
Page No 307:
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7547/NCERT_Solution_Math_Chapter_7_final_html_61e38253.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7547/NCERT_Solution_Math_Chapter_7_final_html_m3ab061dc.gif)
Page No 307:
Question 16:
tan4x
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7549/NCERT_Solution_Math_Chapter_7_final_html_m781709fe.gif)
From equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7549/NCERT_Solution_Math_Chapter_7_final_html_m15daa35b.gif)
Page No 307:
Question 17:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7552/NCERT_Solution_Math_Chapter_7_final_html_390003e3.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7552/NCERT_Solution_Math_Chapter_7_final_html_m4da9ac1c.gif)
Page No 307:
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7554/NCERT_Solution_Math_Chapter_7_final_html_m4c746474.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7554/NCERT_Solution_Math_Chapter_7_final_html_2529779e.gif)
Page No 307:
Question 19:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7557/NCERT_Solution_Math_Chapter_7_final_html_m473f7a6d.gif)
ANSWER:
![](https://ddtarget.com/wp-content/uploads/2024/01/image-62.png)
Page No 307:
Question 20:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7559/NCERT_Solution_Math_Chapter_7_final_html_3ac37af4.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7559/NCERT_Solution_Math_Chapter_7_final_html_m57da292d.gif)
Page No 307:
Question 21:
sin−1 (cos x)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7562/item_82_html_5ca185ec.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7562/item_82_html_34c59b7.gif)
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7562/item_82_html_67f31dd1.gif)
Substituting in equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7562/item_82_html_62bbad50.gif)
Page No 307:
Question 22:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7564/NCERT_Solution_Math_Chapter_7_final_html_m299efddf.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7564/NCERT_Solution_Math_Chapter_7_final_html_m6d95357.gif)
Page No 307:
Question 23:
is equal to
A. tan x + cot x + C
B. tan x + cosec x + C
C. − tan x + cot x + C
D. tan x + sec x + C
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7566/NCERT_Solution_Math_Chapter_7_final_html_m5794483e.gif)
Hence, the correct answer is A.
Page No 307:
Question 24:
equals
A. − cot (exx) + C
B. tan (xex) + C
C. tan (ex) + C
D. cot (ex) + C
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7568/NCERT_Solution_Math_Chapter_7_final_html_m46251d8.gif)
Let exx = t
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7568/NCERT_Solution_Math_Chapter_7_final_html_m410bc16d.gif)
Hence, the correct answer is B.
Page No 315:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7570/NCERT_Solution_Math_Chapter_7_final_html_m6d4a512a.gif)
ANSWER:
Let x3 = t
∴ 3x2dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7570/NCERT_Solution_Math_Chapter_7_final_html_6c319a08.gif)
Page No 315:
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7572/NCERT_Solution_Math_Chapter_7_final_html_m6eadee56.gif)
ANSWER:
Let 2x = t
∴ 2dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7572/NCERT_Solution_Math_Chapter_7_final_html_2bb9ddb6.gif)
Page No 315:
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7574/NCERT_Solution_Math_Chapter_7_final_html_m4f26274f.gif)
ANSWER:
Let 2 − x = t
⇒ −dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7574/NCERT_Solution_Math_Chapter_7_final_html_m311edd7e.gif)
Page No 315:
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7576/NCERT_Solution_Math_Chapter_7_final_html_12ee8f5b.gif)
ANSWER:
Let 5x = t
∴ 5dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7576/NCERT_Solution_Math_Chapter_7_final_html_4152a2c9.gif)
Page No 315:
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7577/NCERT_Solution_Math_Chapter_7_final_html_1313bb5.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7577/NCERT_Solution_Math_Chapter_7_final_html_521d393d.gif)
Page No 315:
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7579/NCERT_Solution_Math_Chapter_7_final_html_686a690e.gif)
ANSWER:
Let x3 = t
∴ 3x2dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7579/NCERT_Solution_Math_Chapter_7_final_html_687cb876.gif)
Page No 315:
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7581/NCERT_Solution_Math_Chapter_7_final_html_m693dd56.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7581/NCERT_Solution_Math_Chapter_7_final_html_m6e073184.gif)
From (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7581/NCERT_Solution_Math_Chapter_7_final_html_m5bd0fa9b.gif)
Page No 315:
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7583/NCERT_Solution_Math_Chapter_7_final_html_1f129e0a.gif)
ANSWER:
Let x3 = t
⇒ 3x2dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7583/NCERT_Solution_Math_Chapter_7_final_html_m163da311.gif)
Page No 315:
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7584/NCERT_Solution_Math_Chapter_7_final_html_m46808fa3.gif)
ANSWER:
Let tan x = t
∴ sec2x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7584/NCERT_Solution_Math_Chapter_7_final_html_654e25c6.gif)
Page No 316:
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7586/NCERT_Solution_Math_Chapter_7_final_html_m94a3620.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7586/NCERT_Solution_Math_Chapter_7_final_html_m518e3476.gif)
Page No 316:
Question 11:
![](https://ddtarget.com/wp-content/uploads/2024/01/image-66.png)
ANSWER:
![](https://ddtarget.com/wp-content/uploads/2024/01/image-65.png)
Page No 316:
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7589/NCERT_Solution_Math_Chapter_7_final_html_168db442.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7589/NCERT_Solution_Math_Chapter_7_final_html_10a5934e.gif)
Page No 316:
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7591/NCERT_Solution_Math_Chapter_7_final_html_62650cce.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7591/NCERT_Solution_Math_Chapter_7_final_html_m7b6ec913.gif)
Page No 316:
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7593/NCERT_Solution_Math_Chapter_7_final_html_62f95dc1.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7593/NCERT_Solution_Math_Chapter_7_final_html_m60221a62.gif)
Page No 316:
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7596/NCERT_Solution_Math_Chapter_7_final_html_m67d6b929.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7596/NCERT_Solution_Math_Chapter_7_final_html_m1d0caab4.gif)
Page No 316:
Question 16:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7598/NCERT_Solution_Math_Chapter_7_final_html_664b5d67.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7598/NCERT_Solution_Math_Chapter_7_final_html_m1b1241f3.gif)
Equating the coefficients of x and constant term on both sides, we obtain
4A = 4 ⇒ A = 1
A + B = 1 ⇒ B = 0
Let 2x2 + x − 3 = t
∴ (4x + 1) dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7598/NCERT_Solution_Math_Chapter_7_final_html_2a4a7359.gif)
Page No 316:
Question 17:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_39c73fad.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_b662a70.gif)
Equating the coefficients of x and constant term on both sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_m668374ed.gif)
From (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_1ddb53d5.gif)
From equation (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7601/NCERT_Solution_Math_Chapter_7_final_html_7731a74.gif)
Page No 316:
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m59186f15.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m72fdaa74.gif)
Equating the coefficient of x and constant term on both sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m1c9032eb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m2bfaa834.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_1ab063c8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m192518e9.gif)
Substituting equations (2) and (3) in equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7607/NCERT_Solution_Math_Chapter_7_final_html_m4ebced4d.gif)
Page No 316:
Question 19:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m9c52b48.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_6a0bbc55.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m5ae3a8ee.gif)
Equating the coefficients of x and constant term, we obtain
2A = 6 ⇒ A = 3
−9A + B = 7 ⇒ B = 34
∴ 6x + 7 = 3 (2x − 9) + 34
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m597fb40f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m55743fdf.gif)
Substituting equations (2) and (3) in (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7612/NCERT_Solution_Math_Chapter_7_final_html_m151359b3.gif)
Page No 316:
Question 20:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_3a973c5a.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_m1e4b7ad4.gif)
Equating the coefficients of x and constant term on both sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_m68eba682.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_17bfe748.gif)
Using equations (2) and (3) in (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7617/NCERT_Solution_Math_Chapter_7_final_html_m70c40c54.gif)
Page No 316:
Question 21:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_4ba941c2.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_m5aaab5ed.gif)
Let x2 + 2x +3 = t
⇒ (2x + 2) dx =dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_1869b74b.gif)
Using equations (2) and (3) in (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_m53d4ecad.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7619/NCERT_Solution_Math_Chapter_7_final_html_m67495a4.gif)
Page No 316:
Question 22:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_m36d7c819.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_7848b5f8.gif)
Equating the coefficients of x and constant term on both sides, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_4b12c12d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_m713d5001.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_68d718b3.gif)
Substituting (2) and (3) in (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7622/NCERT_Solution_Math_Chapter_7_final_html_6aab90c.gif)
Page No 316:
Question 23:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_17e7b92.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_445b6746.gif)
Equating the coefficients of x and constant term, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_m58eeddc7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_m224dadf8.gif)
Using equations (2) and (3) in (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7625/NCERT_Solution_Math_Chapter_7_final_html_m479038b2.gif)
Page No 316:
Question 24:
equals
A. x tan−1 (x + 1) + C
B. tan− 1 (x + 1) + C
C. (x + 1) tan−1x + C
D. tan−1 x + C
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7629/NCERT_Solution_Math_Chapter_7_final_html_m59b21429.gif)
Hence, the correct answer is B.
Page No 316:
Question 25:
equals
A.
B.
C.
![](https://ddtarget.com/wp-content/uploads/2024/01/image-67.png)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7631/NCERT_Solution_Math_Chapter_7_final_html_m563bf5a5.gif)
Hence, the correct answer is B.
Page No 322:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7634/NCERT_Solution_Math_Chapter_7_final_html_m55126bc.gif)
ANSWER:
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7634/NCERT_Solution_Math_Chapter_7_final_html_m16112759.gif)
Equating the coefficients of x and constant term, we obtain
A + B = 1
2A + B = 0
On solving, we obtain
A = −1 and B = 2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7634/NCERT_Solution_Math_Chapter_7_final_html_m4848e6b0.gif)
Page No 322:
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7636/NCERT_Solution_Math_Chapter_7_final_html_m68b00263.gif)
ANSWER:
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7636/NCERT_Solution_Math_Chapter_7_final_html_m28783f7d.gif)
Equating the coefficients of x and constant term, we obtain
A + B = 0
−3A + 3B = 1
On solving, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7636/NCERT_Solution_Math_Chapter_7_final_html_535f7aa7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7636/NCERT_Solution_Math_Chapter_7_final_html_m3d20aa7e.gif)
Page No 322:
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7637/NCERT_Solution_Math_Chapter_7_final_html_47ceee9b.gif)
ANSWER:
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7637/NCERT_Solution_Math_Chapter_7_final_html_26acb356.gif)
Substituting x = 1, 2, and 3 respectively in equation (1), we obtain
A = 1, B = −5, and C = 4
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7637/NCERT_Solution_Math_Chapter_7_final_html_m4450a170.gif)
Page No 322:
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7638/NCERT_Solution_Math_Chapter_7_final_html_m604be94a.gif)
ANSWER:
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7638/NCERT_Solution_Math_Chapter_7_final_html_m1762d17b.gif)
Substituting x = 1, 2, and 3 respectively in equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7638/NCERT_Solution_Math_Chapter_7_final_html_m3e97765a.gif)
Page No 322:
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7640/NCERT_Solution_Math_Chapter_7_final_html_67c72223.gif)
ANSWER:
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7640/NCERT_Solution_Math_Chapter_7_final_html_51425d46.gif)
Substituting x = −1 and −2 in equation (1), we obtain
A = −2 and B = 4
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7640/NCERT_Solution_Math_Chapter_7_final_html_44d773ed.gif)
Page No 322:
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7643/NCERT_Solution_Math_Chapter_7_final_html_m283b87e2.gif)
ANSWER:
It can be seen that the given integrand is not a proper fraction.
Therefore, on dividing (1 − x2) by x(1 − 2x), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7643/NCERT_Solution_Math_Chapter_7_final_html_m6d6cb64.gif)
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7643/NCERT_Solution_Math_Chapter_7_final_html_7abf94a2.gif)
Substituting x = 0 and in equation (1), we obtain
A = 2 and B = 3
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7643/NCERT_Solution_Math_Chapter_7_final_html_m633e2bc7.gif)
Substituting in equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7643/NCERT_Solution_Math_Chapter_7_final_html_m5ef621d.gif)
Page No 322:
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7646/NCERT_Solution_Math_Chapter_7_final_html_4c300254.gif)
ANSWER:
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7646/NCERT_Solution_Math_Chapter_7_final_html_m4f316a86.gif)
Equating the coefficients of x2, x, and constant term, we obtain
A + C = 0
−A + B = 1
−B + C = 0
On solving these equations, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7646/NCERT_Solution_Math_Chapter_7_final_html_1512cd64.gif)
From equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7646/NCERT_Solution_Math_Chapter_7_final_html_m657b90b0.gif)
Page No 322:
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7648/NCERT_Solution_Math_Chapter_7_final_html_33a31e6c.gif)
ANSWER:
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7648/NCERT_Solution_Math_Chapter_7_final_html_4f0f5909.gif)
Substituting x = 1, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7648/NCERT_Solution_Math_Chapter_7_final_html_m16a77e49.gif)
Equating the coefficients of x2 and constant term, we obtain
A + C = 0
−2A + 2B + C = 0
On solving, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7648/NCERT_Solution_Math_Chapter_7_final_html_m620277b2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7648/NCERT_Solution_Math_Chapter_7_final_html_58e6a15b.gif)
Page No 322:
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7649/NCERT_Solution_Math_Chapter_7_final_html_15b962ca.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7649/NCERT_Solution_Math_Chapter_7_final_html_4a16e570.gif)
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7649/NCERT_Solution_Math_Chapter_7_final_html_m7cd0d550.gif)
Substituting x = 1 in equation (1), we obtain
B = 4
Equating the coefficients of x2 and x, we obtain
A + C = 0
B − 2C = 3
On solving, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7649/NCERT_Solution_Math_Chapter_7_final_html_m643a66d1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7649/NCERT_Solution_Math_Chapter_7_final_html_m4e34711f.gif)
Page No 322:
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7651/NCERT_Solution_Math_Chapter_7_final_html_7dfa52ef.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7651/NCERT_Solution_Math_Chapter_7_final_html_mcf89333.gif)
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7651/NCERT_Solution_Math_Chapter_7_final_html_48a2f765.gif)
Equating the coefficients of x2 and x, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7651/NCERT_Solution_Math_Chapter_7_final_html_m4ee3b74.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7651/NCERT_Solution_Math_Chapter_7_final_html_6a680da4.gif)
Page No 322:
Question 11:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7653/NCERT_Solution_Math_Chapter_7_final_html_meef6cee.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7653/NCERT_Solution_Math_Chapter_7_final_html_7f58a1eb.gif)
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7653/NCERT_Solution_Math_Chapter_7_final_html_m3492ed42.gif)
Substituting x = −1, −2, and 2 respectively in equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7653/NCERT_Solution_Math_Chapter_7_final_html_5b116a13.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7653/NCERT_Solution_Math_Chapter_7_final_html_m4654c7c5.gif)
Page No 322:
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7655/NCERT_Solution_Math_Chapter_7_final_html_m7e9bdf6f.gif)
ANSWER:
It can be seen that the given integrand is not a proper fraction.
Therefore, on dividing (x3 + x + 1) by x2 − 1, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7655/NCERT_Solution_Math_Chapter_7_final_html_m70e87107.gif)
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7655/NCERT_Solution_Math_Chapter_7_final_html_m3d304e42.gif)
Substituting x = 1 and −1 in equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7655/NCERT_Solution_Math_Chapter_7_final_html_m117d1cc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7655/NCERT_Solution_Math_Chapter_7_final_html_9b89357.gif)
Page No 322:
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7657/NCERT_Solution_Math_Chapter_7_final_html_1b0b274d.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7657/NCERT_Solution_Math_Chapter_7_final_html_31515503.gif)
Equating the coefficient of x2, x, and constant term, we obtain
A − B = 0
B − C = 0
A + C = 2
On solving these equations, we obtain
A = 1, B = 1, and C = 1
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7657/NCERT_Solution_Math_Chapter_7_final_html_m6a802ca2.gif)
Page No 322:
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7659/NCERT_Solution_Math_Chapter_7_final_html_1d0293db.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7659/NCERT_Solution_Math_Chapter_7_final_html_m151d55ff.gif)
Equating the coefficient of x and constant term, we obtain
A = 3
2A + B = −1 ⇒ B = −7
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7659/NCERT_Solution_Math_Chapter_7_final_html_m46af344a.gif)
Page No 322:
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7662/NCERT_Solution_Math_Chapter_7_final_html_m63e2a732.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7662/NCERT_Solution_Math_Chapter_7_final_html_m23dc4fc1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7662/NCERT_Solution_Math_Chapter_7_final_html_1b491979.gif)
Equating the coefficient of x3, x2, x, and constant term, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7662/NCERT_Solution_Math_Chapter_7_final_html_m712c361.gif)
On solving these equations, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7662/NCERT_Solution_Math_Chapter_7_final_html_m4d8f1b57.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7662/NCERT_Solution_Math_Chapter_7_final_html_a36f335.gif)
Page No 322:
Question 16:
[Hint: multiply numerator and denominator by xn − 1 and put xn = t]
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7663/NCERT_Solution_Math_Chapter_7_final_html_m5f37d30f.gif)
Multiplying numerator and denominator by xn − 1, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7663/NCERT_Solution_Math_Chapter_7_final_html_m202e0f7f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7663/NCERT_Solution_Math_Chapter_7_final_html_m25513d6a.gif)
Substituting t = 0, −1 in equation (1), we obtain
A = 1 and B = −1
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7663/NCERT_Solution_Math_Chapter_7_final_html_m1437c43a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7663/NCERT_Solution_Math_Chapter_7_final_html_m3402fc01.gif)
Page No 322:
Question 17:
[Hint: Put sin x = t]
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7665/NCERT_Solution_Math_Chapter_7_final_html_40e59867.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7665/NCERT_Solution_Math_Chapter_7_final_html_1c3aabf6.gif)
Substituting t = 2 and then t = 1 in equation (1), we obtain
A = 1 and B = −1
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7665/NCERT_Solution_Math_Chapter_7_final_html_1bd280c7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7665/NCERT_Solution_Math_Chapter_7_final_html_m62760a7e.gif)
Page No 323:
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7668/NCERT_Solution_Math_Chapter_7_final_html_63cc9e04.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7668/NCERT_Solution_Math_Chapter_7_final_html_67db95a0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7668/NCERT_Solution_Math_Chapter_7_final_html_14356613.gif)
Equating the coefficients of x3, x2, x, and constant term, we obtain
A + C = 0
B + D = 4
4A + 3C = 0
4B + 3D = 10
On solving these equations, we obtain
A = 0, B = −2, C = 0, and D = 6
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7668/NCERT_Solution_Math_Chapter_7_final_html_473e50ca.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7668/NCERT_Solution_Math_Chapter_7_final_html_591d2419.gif)
Page No 323:
Question 19:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_29c714c8.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_29c714c8.gif)
Let x2 = t ⇒ 2x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_709f9dff.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_m740aa2df.gif)
Substituting t = −3 and t = −1 in equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_693a6469.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_m68da9ac5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7669/NCERT_Solution_Math_Chapter_7_final_html_2c43a007.gif)
Page No 323:
Question 20:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_12c7d30a.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_12c7d30a.gif)
Multiplying numerator and denominator by x3, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_36624ea7.gif)
Let x4 = t ⇒ 4x3dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_6efe54f8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_m40c17f5.gif)
Substituting t = 0 and 1 in (1), we obtain
A = −1 and B = 1
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_5a5b384f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7671/NCERT_Solution_Math_Chapter_7_final_html_m4c782b7a.gif)
Page No 323:
Question 21:
[Hint: Put ex = t]
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7672/NCERT_Solution_Math_Chapter_7_final_html_m15581dd.gif)
Let ex = t ⇒ exdx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7672/NCERT_Solution_Math_Chapter_7_final_html_m5058b23.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7672/NCERT_Solution_Math_Chapter_7_final_html_16b2ac81.gif)
Substituting t = 1 and t = 0 in equation (1), we obtain
A = −1 and B = 1
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7672/NCERT_Solution_Math_Chapter_7_final_html_237dc92f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7672/NCERT_Solution_Math_Chapter_7_final_html_m4e47628f.gif)
Page No 323:
Question 22:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7673/NCERT_Solution_Math_Chapter_7_final_html_m3f6b4c23.gif)
A.
B.
C.
D.
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7673/NCERT_Solution_Math_Chapter_7_final_html_m1a39d92a.gif)
Substituting x = 1 and 2 in (1), we obtain
A = −1 and B = 2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7673/NCERT_Solution_Math_Chapter_7_final_html_m1da97f65.gif)
Hence, the correct answer is B.
Page No 323:
Question 23:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7674/NCERT_Solution_Math_Chapter_7_final_html_3ee528b8.gif)
A.
B.
C.
D.
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7674/NCERT_Solution_Math_Chapter_7_final_html_1c4b44d7.gif)
Equating the coefficients of x2, x, and constant term, we obtain
A + B = 0
C = 0
A = 1
On solving these equations, we obtain
A = 1, B = −1, and C = 0
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7674/NCERT_Solution_Math_Chapter_7_final_html_6b082199.gif)
Hence, the correct answer is A.
Page No 327:
Question 1:
x sin x
ANSWER:
Let I =
Taking x as first function and sin x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7675/NCERT_Solution_Math_Chapter_7_final_html_m5b52456b.gif)
Page No 327:
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7676/NCERT_Solution_Math_Chapter_7_final_html_m565b8df6.gif)
ANSWER:
Let I =
Taking x as first function and sin 3x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7676/NCERT_Solution_Math_Chapter_7_final_html_m214c7ae3.gif)
Page No 327:
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7677/NCERT_Solution_Math_Chapter_7_final_html_45da2d8.gif)
ANSWER:
Let
Taking x2 as first function and ex as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7677/NCERT_Solution_Math_Chapter_7_final_html_m1406435d.gif)
Again integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7677/NCERT_Solution_Math_Chapter_7_final_html_243d97bb.gif)
Page No 327:
Question 4:
x logx
ANSWER:
Let
Taking log x as first function and x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7678/NCERT_Solution_Math_Chapter_7_final_html_m5b95e34d.gif)
Page No 327:
Question 5:
x log 2x
ANSWER:
Let
Taking log 2x as first function and x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7679/NCERT_Solution_Math_Chapter_7_final_html_m50a12fc3.gif)
Page No 327:
Question 6:
x2 log x
ANSWER:
Let
Taking log x as first function and x2 as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7680/NCERT_Solution_Math_Chapter_7_final_html_126181e4.gif)
Page No 327:
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7681/NCERT_Solution_Math_Chapter_7_final_html_m70b6b8fb.gif)
ANSWER:
Let
Taking as first function and x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7681/NCERT_Solution_Math_Chapter_7_final_html_m3a459646.gif)
Page No 327:
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7682/NCERT_Solution_Math_Chapter_7_final_html_11d5bd80.gif)
ANSWER:
Let
Taking as first function and x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7682/NCERT_Solution_Math_Chapter_7_final_html_m19e79690.gif)
Page No 327:
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7683/NCERT_Solution_Math_Chapter_7_final_html_5ad5a2f.gif)
ANSWER:
Let
Taking cos−1 x as first function and x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7683/NCERT_Solution_Math_Chapter_7_final_html_1991c6d7.gif)
Page No 327:
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7684/NCERT_Solution_Math_Chapter_7_final_html_2aa59a77.gif)
ANSWER:
Let
Taking as first function and 1 as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7684/NCERT_Solution_Math_Chapter_7_final_html_783e653d.gif)
Page No 327:
Question 11:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_m7beafb64.gif)
ANSWER:
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_m54c945f9.gif)
Taking as first function and
as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7686/NCERT_Solution_Math_Chapter_7_final_html_56d7e816.gif)
Page No 327:
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7687/NCERT_Solution_Math_Chapter_7_final_html_4e99fa3a.gif)
ANSWER:
Let
Taking x as first function and sec2x as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7687/NCERT_Solution_Math_Chapter_7_final_html_m384594e7.gif)
Page No 327:
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7688/NCERT_Solution_Math_Chapter_7_final_html_m1a8b5181.gif)
ANSWER:
Let
Taking as first function and 1 as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7688/NCERT_Solution_Math_Chapter_7_final_html_m4bf69e04.gif)
Page No 327:
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7689/NCERT_Solution_Math_Chapter_7_final_html_541373dc.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7689/NCERT_Solution_Math_Chapter_7_final_html_m783b9696.gif)
Taking as first function and x as second function and integrating by parts, we obtain
![](https://ddtarget.com/wp-content/uploads/2024/01/image-68.png)
![](https://ddtarget.com/wp-content/uploads/2024/01/image-69.png)
Page No 327:
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m7e288453.gif)
ANSWER:
Let
Let I = I1 + I2 … (1)
Where, and
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_508ac40a.gif)
Taking log x as first function and x2 as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_64161ae9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m3afbd7eb.gif)
Taking log x as first function and 1 as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_m33301466.gif)
Using equations (2) and (3) in (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7691/NCERT_Solution_Math_Chapter_7_final_html_32c47506.gif)
Page No 328:
Question 16:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_m16dbffa1.gif)
ANSWER:
Let
Let
⇒
∴
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7695/NCERT_Solution_Math_Chapter_7_final_html_ma0bf94e.gif)
Page No 328:
Question 17:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_41e8a549.gif)
ANSWER:
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_356f9e57.gif)
Let ⇒
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_m692fc956.gif)
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7697/NCERT_Solution_Math_Chapter_7_final_html_m234a6149.gif)
Page No 328:
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_2d60c21c.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_1b8fb60a.gif)
Let ⇒
It is known that,
From equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7699/NCERT_Solution_Math_Chapter_7_final_html_m483f6657.gif)
Page No 328:
Question 19:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_425f49ab.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_509a61e0.gif)
Also, let ⇒
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7702/NCERT_Solution_Math_Chapter_7_final_html_m7a0cc9fc.gif)
Page No 328:
Question 20:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_m5d04b491.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_m1ffb9b2d.gif)
Let ⇒
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7705/NCERT_Solution_Math_Chapter_7_final_html_m3379bca.gif)
Page No 328:
Question 21:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7707/NCERT_Solution_Math_Chapter_7_final_html_m198871a2.gif)
ANSWER:
Let
Integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7707/NCERT_Solution_Math_Chapter_7_final_html_m72845c80.gif)
Again integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7707/NCERT_Solution_Math_Chapter_7_final_html_m624ee5cc.gif)
Page No 328:
Question 22:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_5c4b1fa6.gif)
ANSWER:
Let ⇒
= 2θ
⇒
Integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7710/NCERT_Solution_Math_Chapter_7_final_html_m2f12cf1c.gif)
Page No 328:
Question 23:
equals
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_m2073483f.gif)
ANSWER:
Let
Also, let ⇒
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7712/NCERT_Solution_Math_Chapter_7_final_html_m3e91749d.gif)
Hence, the correct answer is A.
Page No 328:
Question 24:
equals
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_42378067.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_46f2088a.gif)
Let
Also, let ⇒
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7714/NCERT_Solution_Math_Chapter_7_final_html_31dddd75.gif)
Hence, the correct answer is B.
Page No 330:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7716/NCERT_Solution_Math_Chapter_7_final_html_m1170ca09.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_14_13_39/NCERT_Solution_Math_Chapter_7_final_html_55b76281_1748512521494652501.png)
![](https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_14_13_39/NCERT_Solution_Math_Chapter_7_final_html_55b76281_4482949128659630918.png)
![](https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_14_13_39/NCERT_Solution_Math_Chapter_7_final_html_55b76281_6819830928611292475.png)
![](https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_14_13_39/NCERT_Solution_Math_Chapter_7_final_html_55b76281_5766645889198152699.png)
![](https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_14_13_39/NCERT_Solution_Math_Chapter_7_final_html_55b76281_2274410893609283042.png)
Page No 330:
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7719/NCERT_Solution_Math_Chapter_7_final_html_m524978b7.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7719/NCERT_Solution_Math_Chapter_7_final_html_m25d68940.gif)
Page No 330:
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7721/NCERT_Solution_Math_Chapter_7_final_html_1078b016.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7721/NCERT_Solution_Math_Chapter_7_final_html_12ee79a6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7721/NCERT_Solution_Math_Chapter_7_final_html_3dc605c1.gif)
Page No 330:
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7722/NCERT_Solution_Math_Chapter_7_final_html_m2833efcd.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7722/NCERT_Solution_Math_Chapter_7_final_html_6c3bdbca.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7722/NCERT_Solution_Math_Chapter_7_final_html_m5cf48318.gif)
Page No 330:
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7724/NCERT_Solution_Math_Chapter_7_final_html_4046b156.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7724/NCERT_Solution_Math_Chapter_7_final_html_3fe6d1f5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7724/NCERT_Solution_Math_Chapter_7_final_html_49e1f051.gif)
Page No 330:
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7725/NCERT_Solution_Math_Chapter_7_final_html_m54dece65.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7725/NCERT_Solution_Math_Chapter_7_final_html_7cc40572.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7725/NCERT_Solution_Math_Chapter_7_final_html_482a9a24.gif)
Page No 330:
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7727/NCERT_Solution_Math_Chapter_7_final_html_m29dd6f5d.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7727/NCERT_Solution_Math_Chapter_7_final_html_m2460c660.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7727/NCERT_Solution_Math_Chapter_7_final_html_m5fffe454.gif)
Page No 330:
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7729/NCERT_Solution_Math_Chapter_7_final_html_m19083a3b.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7729/NCERT_Solution_Math_Chapter_7_final_html_m3e6f3b43.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7729/NCERT_Solution_Math_Chapter_7_final_html_m16dfab03.gif)
Page No 330:
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7731/NCERT_Solution_Math_Chapter_7_final_html_191c58cb.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7731/NCERT_Solution_Math_Chapter_7_final_html_m3f7229fc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7731/NCERT_Solution_Math_Chapter_7_final_html_m5f522660.gif)
Page No 330:
Question 10:
is equal to
A.
B.
C.
![](https://ddtarget.com/wp-content/uploads/2024/01/image-70.png)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7732/NCERT_Solution_Math_Chapter_7_final_html_m6520b32b.gif)
Hence, the correct answer is A.
Page No 330:
Question 11:
is equal to
A.
B.
C.
D.
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7733/NCERT_Solution_Math_Chapter_7_final_html_56279587.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7733/NCERT_Solution_Math_Chapter_7_final_html_m32d28a69.gif)
Hence, the correct answer is D.
Page No 334:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7734/NCERT_Solution_Math_Chapter_7_final_html_69e5f3b9.gif)
ANSWER:
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7734/NCERT_Solution_Math_Chapter_7_final_html_me7fec4d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7734/NCERT_Solution_Math_Chapter_7_final_html_72e7930a.gif)
Page No 334:
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7735/NCERT_Solution_Math_Chapter_7_final_html_m7ea19f1b.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7735/NCERT_Solution_Math_Chapter_7_final_html_5fe9d3a4.gif)
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7735/NCERT_Solution_Math_Chapter_7_final_html_m405877a1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7735/NCERT_Solution_Math_Chapter_7_final_html_446ceb1.gif)
Page No 334:
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7736/NCERT_Solution_Math_Chapter_7_final_html_m360250ef.gif)
ANSWER:
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7736/NCERT_Solution_Math_Chapter_7_final_html_m4f945515.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7736/NCERT_Solution_Math_Chapter_7_final_html_2aa04a39.gif)
Page No 334:
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_m220fbce.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_m2879635d.gif)
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_m383850e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_m6bd88959.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_5952197b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_47ebd74.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_51a9ceb8.gif)
From equations (2) and (3), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7737/NCERT_Solution_Math_Chapter_7_final_html_1bb6734a.gif)
Page No 334:
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7739/NCERT_Solution_Math_Chapter_7_final_html_m1cc03f8e.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7739/NCERT_Solution_Math_Chapter_7_final_html_57b2e480.gif)
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7739/NCERT_Solution_Math_Chapter_7_final_html_m4ddc80d9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7739/NCERT_Solution_Math_Chapter_7_final_html_7c5270a4.gif)
Page No 334:
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7741/NCERT_Solution_Math_Chapter_7_final_html_2690150b.gif)
ANSWER:
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7741/NCERT_Solution_Math_Chapter_7_final_html_m23e187a5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7741/NCERT_Solution_Math_Chapter_7_final_html_164cb7ca.gif)
Page No 338:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7742/NCERT_Solution_Math_Chapter_7_final_html_m1067703e.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7742/NCERT_Solution_Math_Chapter_7_final_html_39473b81.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7742/NCERT_Solution_Math_Chapter_7_final_html_m5b213be6.gif)
Page No 338:
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7743/NCERT_Solution_Math_Chapter_7_final_html_480f9557.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7743/NCERT_Solution_Math_Chapter_7_final_html_51690e6.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7743/NCERT_Solution_Math_Chapter_7_final_html_303663d0.gif)
Page No 338:
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7745/NCERT_Solution_Math_Chapter_7_final_html_m70152f9b.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7745/NCERT_Solution_Math_Chapter_7_final_html_m20e8f6a9.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7745/NCERT_Solution_Math_Chapter_7_final_html_m35238a09.gif)
Page No 338:
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7746/NCERT_Solution_Math_Chapter_7_final_html_4ba6d54a.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7746/NCERT_Solution_Math_Chapter_7_final_html_m2e6c9f5f.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7746/NCERT_Solution_Math_Chapter_7_final_html_6256f83d.gif)
Page No 338:
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7747/Items%20179_html_1c52a724.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7747/Items%20179_html_608a452e.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7747/Items%20179_html_m3cd3c92d.gif)
Page No 338:
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7749/NCERT_Solution_Math_Chapter_7_final_html_426fe2e4.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7749/NCERT_Solution_Math_Chapter_7_final_html_m5c2222aa.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7749/NCERT_Solution_Math_Chapter_7_final_html_m6026a9a1.gif)
Page No 338:
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7751/NCERT_Solution_Math_Chapter_7_final_html_m38d7921f.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7751/NCERT_Solution_Math_Chapter_7_final_html_m4d2e837b.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7751/NCERT_Solution_Math_Chapter_7_final_html_m44340047.gif)
Page No 338:
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7752/NCERT_Solution_Math_Chapter_7_final_html_m1498ff3f.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7752/NCERT_Solution_Math_Chapter_7_final_html_m6da61a7.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7752/NCERT_Solution_Math_Chapter_7_final_html_m7c581a2d.gif)
Page No 338:
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7754/NCERT_Solution_Math_Chapter_7_final_html_m166e5b62.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7754/NCERT_Solution_Math_Chapter_7_final_html_m2b080397.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7754/NCERT_Solution_Math_Chapter_7_final_html_14d81098.gif)
Page No 338:
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7756/NCERT_Solution_Math_Chapter_7_final_html_m81460a9.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7756/NCERT_Solution_Math_Chapter_7_final_html_m4f2ac16.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7756/NCERT_Solution_Math_Chapter_7_final_html_m2c4074b2.gif)
Page No 338:
Question 11:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7760/NCERT_Solution_Math_Chapter_7_final_html_682faf17.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7760/NCERT_Solution_Math_Chapter_7_final_html_61f2f3f1.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7760/NCERT_Solution_Math_Chapter_7_final_html_m2f2f6fbd.gif)
Page No 338:
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7762/Item%20186_html_407b64a3.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7762/Item%20186_html_7c21c7f5.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7762/Item%20186_html_m19730f06.gif)
Page No 338:
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7763/NCERT_Solution_Math_Chapter_7_final_html_79211595.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7763/NCERT_Solution_Math_Chapter_7_final_html_m9bd612c.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7763/NCERT_Solution_Math_Chapter_7_final_html_m45040b1.gif)
Page No 338:
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7764/NCERT_Solution_Math_Chapter_7_final_html_1c0733f6.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7764/NCERT_Solution_Math_Chapter_7_final_html_7b54546e.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7764/NCERT_Solution_Math_Chapter_7_final_html_mebefe21.gif)
Page No 338:
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7765/NCERT_Solution_Math_Chapter_7_final_html_6d91cb30.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7765/NCERT_Solution_Math_Chapter_7_final_html_m6ca16809.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7765/NCERT_Solution_Math_Chapter_7_final_html_m2160b3b9.gif)
Page No 338:
Question 16:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_1a209d0c.gif)
ANSWER:
Let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_m3e58e905.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_2453d8c5.gif)
Equating the coefficients of x and constant term, we obtain
A = 10 and B = −25
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_91e9a9d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_90155e9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_m9471e1f.gif)
Substituting the value of I1 in (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7766/NCERT_Solution_Math_Chapter_7_final_html_m6af71c41.gif)
Page No 338:
Question 17:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7767/NCERT_Solution_Math_Chapter_7_final_html_62748c4f.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7767/NCERT_Solution_Math_Chapter_7_final_html_799ad7cb.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7767/NCERT_Solution_Math_Chapter_7_final_html_m160b5348.gif)
Page No 338:
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7769/Item%20192_html_m1a378172.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7769/Item%20192_html_3225002c.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7769/Item%20192_html_m58ae84a8.gif)
Page No 338:
Question 19:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7771/Item%20193_html_24e76ff5.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7771/Item%20193_html_6ea94b8.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7771/Item%20193_html_17842141.gif)
Page No 338:
Question 20:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7774/Item%20194_html_m53e235e1.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7774/Item%20194_html_662f5233.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7774/Item%20194_html_1dd66a6f.gif)
Page No 338:
Question 21:
equals
A.
B.
C.
D.
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7777/NCERT_Solution_Math_Chapter_7_final_html_m4a2a6a82.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7777/NCERT_Solution_Math_Chapter_7_final_html_m6107f62f.gif)
Hence, the correct answer is D.
Page No 338:
Question 22:
equals
A.
B.
C.
D.
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7780/Item%20196_html_39a413bc.gif)
By second fundamental theorem of calculus, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7780/Item%20196_html_3ccb1065.gif)
Hence, the correct answer is C.
Page No 340:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7784/NCERT_Solution_Math_Chapter_7_final_html_6d4f200d.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7784/NCERT_Solution_Math_Chapter_7_final_html_26c4315c.gif)
When x = 0, t = 1 and when x = 1, t = 2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7784/NCERT_Solution_Math_Chapter_7_final_html_mb530b8a.gif)
Page No 340:
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7786/NCERT_Solution_Math_Chapter_7_final_html_m4c0d7ecb.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7786/NCERT_Solution_Math_Chapter_7_final_html_m4f40448a.gif)
Also, let
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7786/NCERT_Solution_Math_Chapter_7_final_html_e432d87.gif)
Page No 340:
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_m37dfca2b.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_m5f5ba4a2.gif)
Also, let x = tanθ ⇒ dx = sec2θ dθ
When x = 0, θ = 0 and when x = 1,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_7416ad9b.gif)
Takingθas first function and sec2θ as second function and integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7789/NCERT_Solution_Math_Chapter_7_final_html_m1696ceaa.gif)
Page No 340:
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7792/NCERT_Solution_Math_Chapter_7_final_html_m6b4ab1ea.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7792/NCERT_Solution_Math_Chapter_7_final_html_m4fa51e4.gif)
Let x + 2 = t2 ⇒ dx = 2tdt
When x = 0, and when x = 2, t = 2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7792/NCERT_Solution_Math_Chapter_7_final_html_671afa1.gif)
Page No 340:
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7793/NCERT_Solution_Math_Chapter_7_final_html_m21d4da58.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7793/NCERT_Solution_Math_Chapter_7_final_html_m21d4da58.gif)
Let cos x = t ⇒ −sinx dx = dt
When x = 0, t = 1 and when
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7793/NCERT_Solution_Math_Chapter_7_final_html_7c4db91c.gif)
Page No 340:
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7796/NCERT_Solution_Math_Chapter_7_final_html_m47f8aec.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7796/NCERT_Solution_Math_Chapter_7_final_html_m7f64c05b.gif)
Let ⇒ dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7796/NCERT_Solution_Math_Chapter_7_final_html_6b4b32a9.gif)
Page No 340:
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7798/Item%20203_html_m6af4a309.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7798/Item%20203_html_37652a3b.gif)
Let x + 1 = t ⇒ dx = dt
When x = −1, t = 0 and when x = 1, t = 2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7798/Item%20203_html_5fb9abe8.gif)
Page No 340:
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7800/NCERT_Solution_Math_Chapter_7_final_html_6979ef1b.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7800/NCERT_Solution_Math_Chapter_7_final_html_6979ef1b.gif)
Let 2x = t ⇒ 2dx = dt
When x = 1, t = 2 and when x = 2, t = 4
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7800/NCERT_Solution_Math_Chapter_7_final_html_71eddcf4.gif)
Page No 340:
Question 9:
The value of the integral is
A. 6
B. 0
C. 3
D. 4
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7802/Item%20205_html_590694d8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7802/Item%20205_html_61485027.gif)
Let cot θ = t ⇒ −cosec2 θdθ= dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7802/Item%20205_html_1d3dd9f2.gif)
Hence, the correct answer is A.
Page No 340:
Question 10:
If
A. cos x + x sin x
B. x sin x
C. x cos x
D. sin x + x cos x
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7806/NCERT_Solution_Math_Chapter_7_final_html_4bad8c8f.gif)
Integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7806/NCERT_Solution_Math_Chapter_7_final_html_m7c5c373f.gif)
Hence, the correct answer is B.
Page No 347:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8321/NCERT_Solution_Math_Chapter_7_final_html_407b64a3.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8321/NCERT_Solution_Math_Chapter_7_final_html_7f54b990.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8321/NCERT_Solution_Math_Chapter_7_final_html_319681a6.gif)
Page No 347:
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8323/NCERT_Solution_Math_Chapter_7_final_html_54ab9afb.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8323/NCERT_Solution_Math_Chapter_7_final_html_m67aa2b1f.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8323/NCERT_Solution_Math_Chapter_7_final_html_664829e1.gif)
Page No 347:
Question 3:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8326/NCERT_Solution_Math_Chapter_7_final_html_m423eccee.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8326/NCERT_Solution_Math_Chapter_7_final_html_35986860.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8326/NCERT_Solution_Math_Chapter_7_final_html_5ec97145.gif)
Page No 347:
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8327/NCERT_Solution_Math_Chapter_7_final_html_4a4c489.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8327/NCERT_Solution_Math_Chapter_7_final_html_32989753.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8327/NCERT_Solution_Math_Chapter_7_final_html_21644107.gif)
Page No 347:
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8328/NCERT_Solution_Math_Chapter_7_final_html_m5de665ca.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8328/NCERT_Solution_Math_Chapter_7_final_html_5dbe20ae.gif)
It can be seen that (x + 2) ≤ 0 on [−5, −2] and (x + 2) ≥ 0 on [−2, 5].
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8328/NCERT_Solution_Math_Chapter_7_final_html_773d8ee3.gif)
Page No 347:
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8329/NCERT_Solution_Math_Chapter_7_final_html_6ad4c7e1.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8329/NCERT_Solution_Math_Chapter_7_final_html_m9a4fcf0.gif)
It can be seen that (x − 5) ≤ 0 on [2, 5] and (x − 5) ≥ 0 on [5, 8].
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8329/NCERT_Solution_Math_Chapter_7_final_html_m20ebff4.gif)
Page No 347:
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8330/NCERT_Solution_Math_Chapter_7_final_html_m3484e968.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8330/NCERT_Solution_Math_Chapter_7_final_html_478c6707.gif)
Page No 347:
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8331/NCERT_Solution_Math_Chapter_7_final_html_6c69f563.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8331/NCERT_Solution_Math_Chapter_7_final_html_m23f2161a.gif)
Page No 347:
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8332/NCERT_Solution_Math_Chapter_7_final_html_d58f318.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8332/NCERT_Solution_Math_Chapter_7_final_html_m26b91062.gif)
Page No 347:
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8334/NCERT_Solution_Math_Chapter_7_final_html_m236c3cc6.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8334/NCERT_Solution_Math_Chapter_7_final_html_3eb02be4.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8334/NCERT_Solution_Math_Chapter_7_final_html_32268a59.gif)
Page No 347:
Question 11:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8335/NCERT_Solution_Math_Chapter_7_final_html_66e84c6b.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8335/NCERT_Solution_Math_Chapter_7_final_html_mca9ceae.gif)
As sin2 (−x) = (sin (−x))2 = (−sin x)2 = sin2x, therefore, sin2x is an even function.
It is known that if f(x) is an even function, then
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8335/NCERT_Solution_Math_Chapter_7_final_html_m1f03cb4.gif)
Page No 347:
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8338/Item%20218_html_m4f8e3f37.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8338/Item%20218_html_m5ab190b8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8338/Item%20218_html_m102ed4fd.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8338/Item%20218_html_3be073f7.gif)
Page No 347:
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8339/NCERT_Solution_Math_Chapter_7_final_html_1000a7cf.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8339/NCERT_Solution_Math_Chapter_7_final_html_37730826.gif)
As sin7 (−x) = (sin (−x))7 = (−sin x)7 = −sin7x, therefore, sin2x is an odd function.
It is known that, if f(x) is an odd function, then
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8339/NCERT_Solution_Math_Chapter_7_final_html_m27c8fca6.gif)
Page No 347:
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m53f198ae.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m2ee739fb.gif)
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m6cbd1067.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m18b59200.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8340/item%20220_html_m1b0fe3a3.gif)
Page No 347:
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8341/NCERT_Solution_Math_Chapter_7_final_html_mb6d95c9.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8341/NCERT_Solution_Math_Chapter_7_final_html_1787c899.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8341/NCERT_Solution_Math_Chapter_7_final_html_m26704fd1.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8341/NCERT_Solution_Math_Chapter_7_final_html_42f38cbe.gif)
Page No 347:
Question 16:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_2c1b6167.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_m17c379d8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_2d540b86.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_6ba7be51.gif)
sin (π − x) = sin x
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_71e40f52.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_5a0010e1.gif)
Adding (4) and (5), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8320/items_222_html_2580c48b.gif)
Let 2x = t ⇒ 2dx = dt
![](https://ddtarget.com/wp-content/uploads/2024/01/image-73.png)
Page No 347:
Question 17:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_2ebf4e1d.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_m17123bbd.gif)
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_m67347065.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8322/NCERT_Solution_Math_Chapter_7_final_html_7f16d75.gif)
Page No 347:
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8324/NCERT_Solution_Math_Chapter_7_final_html_m61e0a3ca.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8324/NCERT_Solution_Math_Chapter_7_final_html_m7281926a.gif)
It can be seen that, (x − 1) ≤ 0 when 0 ≤ x ≤ 1 and (x − 1) ≥ 0 when 1 ≤ x ≤ 4
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8324/NCERT_Solution_Math_Chapter_7_final_html_m41aba784.gif)
Page No 347:
Question 19:
Show that if f and g are defined as
and
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_3bb51308.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_m34610f85.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8325/NCERT_Solution_Math_Chapter_7_final_html_m572b38bb.gif)
Page No 347:
Question 20:
The value of is
A. 0
B. 2
C. π
D. 1
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8333/NCERT_Solution_Math_Chapter_7_final_html_1470d63d.gif)
It is known that if f(x) is an even function, then and
if f(x) is an odd function, then
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8333/NCERT_Solution_Math_Chapter_7_final_html_m125c2688.gif)
Hence, the correct answer is C.
Page No 347:
Question 21:
The value of is
A. 2
B.
C. 0
D.
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_30dd75fb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_m28885fd1.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8336/NCERT_Solution_Math_Chapter_7_final_html_m42a37c68.gif)
Hence, the correct answer is C.
Page No 352:
Question 1:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8337/NCERT_Solution_Math_Chapter_7_final_html_m76d580f5.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8337/NCERT_Solution_Math_Chapter_7_final_html_1cd7312f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8337/NCERT_Solution_Math_Chapter_7_final_html_m5f5406d1.gif)
Equating the coefficients of x2, x, and constant term, we obtain
−A + B − C = 0
B + C = 0
A = 1
On solving these equations, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8337/NCERT_Solution_Math_Chapter_7_final_html_m3de7c279.gif)
From equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/8337/NCERT_Solution_Math_Chapter_7_final_html_m578c8f06.gif)
Page No 352:
Question 2:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7953/NCERT_Solution_Math_Chapter_7_final_html_m1df77264.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7953/NCERT_Solution_Math_Chapter_7_final_html_m4999ef5e.gif)
Page No 352:
Question 3:
[Hint: Put
]
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7947/NCERT_Solution_Math_Chapter_7_final_html_5ade6d76.gif)
Page No 352:
Question 4:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7948/NCERT_Solution_Math_Chapter_7_final_html_5dacd89f.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7948/NCERT_Solution_Math_Chapter_7_final_html_256617de.gif)
Page No 352:
Question 5:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7949/NCERT_Solution_Math_Chapter_7_final_html_m1c750f15.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7949/NCERT_Solution_Math_Chapter_7_final_html_4257a166.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7949/NCERT_Solution_Math_Chapter_7_final_html_m61d78b39.gif)
On dividing, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7949/NCERT_Solution_Math_Chapter_7_final_html_2b0e959c.gif)
Page No 352:
Question 6:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7950/NCERT_Solution_Math_Chapter_7_final_html_77c6b083.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7950/NCERT_Solution_Math_Chapter_7_final_html_m1d1e21a2.gif)
Equating the coefficients of x2, x, and constant term, we obtain
A + B = 0
B + C = 5
9A + C = 0
On solving these equations, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7950/NCERT_Solution_Math_Chapter_7_final_html_349422a5.gif)
From equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7950/NCERT_Solution_Math_Chapter_7_final_html_578f8b8e.gif)
Page No 352:
Question 7:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7951/NCERT_Solution_Math_Chapter_7_final_html_m1249c9d6.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7951/NCERT_Solution_Math_Chapter_7_final_html_m1249c9d6.gif)
Let x − a = t ⇒ dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7951/NCERT_Solution_Math_Chapter_7_final_html_2b521cac.gif)
Page No 352:
Question 8:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7952/NCERT_Solution_Math_Chapter_7_final_html_40aaf85b.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7952/NCERT_Solution_Math_Chapter_7_final_html_471dd3f.gif)
Page No 352:
Question 9:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7918/NCERT_Solution_Math_Chapter_7_final_html_m3fc14e9.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7918/NCERT_Solution_Math_Chapter_7_final_html_m3fc14e9.gif)
Let sin x = t ⇒ cos x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7918/NCERT_Solution_Math_Chapter_7_final_html_m6cb4b391.gif)
Page No 352:
Question 10:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7922/NCERT_Solution_Math_Chapter_7_final_html_m483003f0.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7922/NCERT_Solution_Math_Chapter_7_final_html_474a8251.gif)
Page No 352:
Question 11:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7924/NCERT_Solution_Math_Chapter_7_final_html_m52b54278.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7924/NCERT_Solution_Math_Chapter_7_final_html_m50b84c2a.gif)
Page No 352:
Question 12:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7928/NCERT_Solution_Math_Chapter_7_final_html_m735afaad.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7928/NCERT_Solution_Math_Chapter_7_final_html_m735afaad.gif)
Let x4 = t ⇒ 4x3dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7928/NCERT_Solution_Math_Chapter_7_final_html_m6d3e2293.gif)
Page No 352:
Question 13:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7930/NCERT_Solution_Math_Chapter_7_final_html_2df2b31c.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7930/NCERT_Solution_Math_Chapter_7_final_html_2df2b31c.gif)
Let ex = t ⇒ exdx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7930/NCERT_Solution_Math_Chapter_7_final_html_36b2aedd.gif)
Page No 352:
Question 14:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7954/NCERT_Solution_Math_Chapter_7_final_html_1cd0595e.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7954/NCERT_Solution_Math_Chapter_7_final_html_1e1a0b82.gif)
Equating the coefficients of x3, x2, x, and constant term, we obtain
A + C = 0
B + D = 0
4A + C = 0
4B + D = 1
On solving these equations, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7954/NCERT_Solution_Math_Chapter_7_final_html_m35f98eec.gif)
From equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7954/NCERT_Solution_Math_Chapter_7_final_html_m3c6c133d.gif)
Page No 352:
Question 15:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7955/NCERT_Solution_Math_Chapter_7_final_html_m4a699e43.gif)
ANSWER:
= cos3x × sin x
![](https://ddtarget.com/wp-content/uploads/2024/01/image-72.png)
Page No 352:
Question 16:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7956/NCERT_Solution_Math_Chapter_7_final_html_m62ecd5df.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7956/NCERT_Solution_Math_Chapter_7_final_html_m79262ed9.gif)
Page No 352:
Question 17:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7957/NCERT_Solution_Math_Chapter_7_final_html_m12636567.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7957/NCERT_Solution_Math_Chapter_7_final_html_1c9df5fd.gif)
Page No 352:
Question 18:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7958/NCERT_Solution_Math_Chapter_7_final_html_fc4380a.gif)
ANSWER:
![](https://ddtarget.com/wp-content/uploads/2024/01/image-71.png)
Page No 352:
Question 19:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7959/NCERT_Solution_Math_Chapter_7_final_html_m633ccb7.gif)
ANSWER:
![](https://ddtarget.com/wp-content/uploads/2024/01/image-75.png)
![](https://ddtarget.com/wp-content/uploads/2024/01/image-74.png)
Page No 352:
Question 20:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7960/NCERT_Solution_Math_Chapter_7_final_html_50604b7.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7960/NCERT_Solution_Math_Chapter_7_final_html_762a0d1a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7960/NCERT_Solution_Math_Chapter_7_final_html_3e40ed34.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7960/NCERT_Solution_Math_Chapter_7_final_html_m1c3fbf3.gif)
Page No 352:
Question 21:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7961/NCERT_Solution_Math_Chapter_7_final_html_m4b469c28.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7961/NCERT_Solution_Math_Chapter_7_final_html_2b1a77ef.gif)
Page No 352:
Question 22:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7962/NCERT_Solution_Math_Chapter_7_final_html_69559a26.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7962/NCERT_Solution_Math_Chapter_7_final_html_62dfc439.gif)
Equating the coefficients of x2, x,and constant term, we obtain
A + C = 1
3A + B + 2C = 1
2A + 2B + C = 1
On solving these equations, we obtain
A = −2, B = 1, and C = 3
From equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7962/NCERT_Solution_Math_Chapter_7_final_html_m6aa0e950.gif)
Page No 353:
Question 23:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7963/NCERT_Solution_Math_Chapter_7_final_html_m5796614.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7963/NCERT_Solution_Math_Chapter_7_final_html_m2e79a1c8.gif)
Page No 353:
Question 24:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7964/NCERT_Solution_Math_Chapter_7_final_html_m7bddd4ea.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7964/NCERT_Solution_Math_Chapter_7_final_html_m56d31346.gif)
Integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7964/NCERT_Solution_Math_Chapter_7_final_html_m636f9379.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7964/NCERT_Solution_Math_Chapter_7_final_html_49b75c6e.gif)
Page No 353:
Question 25:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7965/NCERT_Solution_Math_Chapter_7_final_html_m54750f4f.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7965/NCERT_Solution_Math_Chapter_7_final_html_37560177.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7965/NCERT_Solution_Math_Chapter_7_final_html_33bd321a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7965/NCERT_Solution_Math_Chapter_7_final_html_142ee072.gif)
Page No 353:
Question 26:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7966/Items%20253_html_3a7081af.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7966/Items%20253_html_m59f605f1.gif)
When x = 0, t = 0 and
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7966/Items%20253_html_4557600.gif)
Page No 353:
Question 27:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7967/NCERT_Solution_Math_Chapter_7_final_html_m4683c3a.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7967/NCERT_Solution_Math_Chapter_7_final_html_4d9d851e.gif)
When and when
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7967/NCERT_Solution_Math_Chapter_7_final_html_4e934ca9.gif)
Page No 353:
Question 28:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7968/NCERT_Solution_Math_Chapter_7_final_html_m4441a09b.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7968/NCERT_Solution_Math_Chapter_7_final_html_m7484db7e.gif)
When and when
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7968/NCERT_Solution_Math_Chapter_7_final_html_480ad49.gif)
As , therefore,
is an even function.
It is known that if f(x) is an even function, then
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7968/NCERT_Solution_Math_Chapter_7_final_html_m48c28a40.gif)
Page No 353:
Question 29:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7969/NCERT_Solution_Math_Chapter_7_final_html_m776c0f8a.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7969/NCERT_Solution_Math_Chapter_7_final_html_m2b006753.gif)
Page No 353:
Question 30:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7970/NCERT_Solution_Math_Chapter_7_final_html_m52d730c.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7970/NCERT_Solution_Math_Chapter_7_final_html_177cac27.gif)
Page No 353:
Question 31:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7971/Items%20258_html_m33842e5c.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7971/Items%20258_html_m75708a03.gif)
From equation (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7971/Items%20258_html_65374166.gif)
Page No 353:
Question 32:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7972/Items%20259_html_m2f373ec4.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7972/Items%20259_html_30ba894a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7972/Items%20259_html_m4e0d6c68.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7972/Items%20259_html_328fcc21.gif)
Page No 353:
Question 33:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7973/NCERT_Solution_Math_Chapter_7_final_html_mae4faef.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7973/NCERT_Solution_Math_Chapter_7_final_html_21bf0800.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7973/NCERT_Solution_Math_Chapter_7_final_html_m1ab518e9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7973/NCERT_Solution_Math_Chapter_7_final_html_m2dc72d4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7973/NCERT_Solution_Math_Chapter_7_final_html_37eba167.gif)
From equations (1), (2), (3), and (4), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7973/NCERT_Solution_Math_Chapter_7_final_html_m5ef1a4ac.gif)
Page No 353:
Question 34:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7974/NCERT_Solution_Math_Chapter_7_final_html_m48898861.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7974/NCERT_Solution_Math_Chapter_7_final_html_219c8b36.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7974/NCERT_Solution_Math_Chapter_7_final_html_6c9a2099.gif)
Equating the coefficients of x2, x, and constant term, we obtain
A + C = 0
A + B = 0
B = 1
On solving these equations, we obtain
A = −1, C = 1, and B = 1
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7974/NCERT_Solution_Math_Chapter_7_final_html_4c7a449f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7974/NCERT_Solution_Math_Chapter_7_final_html_m336ec5df.gif)
Hence, the given result is proved.
Page No 353:
Question 35:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7975/NCERT_Solution_Math_Chapter_7_final_html_1dd3b058.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7975/NCERT_Solution_Math_Chapter_7_final_html_m202c3c87.gif)
Integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7975/NCERT_Solution_Math_Chapter_7_final_html_6e440035.gif)
Hence, the given result is proved.
Page No 353:
Question 36:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7977/NCERT_Solution_Math_Chapter_7_final_html_7f84818e.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7977/NCERT_Solution_Math_Chapter_7_final_html_678e1db4.gif)
Therefore, f (x) is an odd function.
It is known that if f(x) is an odd function, then
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7977/NCERT_Solution_Math_Chapter_7_final_html_m7faee91f.gif)
Hence, the given result is proved.
Page No 353:
Question 37:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7978/NCERT_Solution_Math_Chapter_7_final_html_41b38be1.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7978/NCERT_Solution_Math_Chapter_7_final_html_148e804.gif)
Hence, the given result is proved.
Page No 353:
Question 38:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7979/NCERT_Solution_Math_Chapter_7_final_html_m53d9d2bb.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7979/NCERT_Solution_Math_Chapter_7_final_html_m1711030b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7979/NCERT_Solution_Math_Chapter_7_final_html_3fb92cfd.gif)
Hence, the given result is proved.
Page No 353:
Question 39:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7980/NCERT_Solution_Math_Chapter_7_final_html_54483698.gif)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7980/NCERT_Solution_Math_Chapter_7_final_html_m23aef4eb.gif)
Integrating by parts, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7980/NCERT_Solution_Math_Chapter_7_final_html_7ba2d404.gif)
Let 1 − x2 = t ⇒ −2x dx = dt
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7980/NCERT_Solution_Math_Chapter_7_final_html_m264f525f.gif)
Hence, the given result is proved.
Page No 353:
Question 40:
Evaluate as a limit of a sum.
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_m1e3a7643.gif)
It is known that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_4cd0ad5d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_1f641d48.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_724206cf.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_743088e5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_42c5fb6f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7981/NCERT_Solution_Math_Chapter_7_final_html_m1d159455.gif)
Page No 353:
Question 41:
is equal to
A.
B.
C.
![](https://ddtarget.com/wp-content/uploads/2024/01/image-79.png)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7982/NCERT_Solution_Math_Chapter_7_final_html_m6eb6c002.gif)
Hence, the correct answer is A.
Page No 353:
Question 42:
is equal to
A.
B.
C.
![](https://ddtarget.com/wp-content/uploads/2024/01/image-78.png)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7943/NCERT_Solution_Math_Chapter_7_final_html_m28e59cdb.gif)
Hence, the correct answer is B.
Page No 354:
Question 43:
If then
is equal to
A.
B.
C.
![](https://ddtarget.com/wp-content/uploads/2024/01/image-77.png)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7939/NCERT_Solution_Math_Chapter_7_final_html_m6cc05e7b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7939/NCERT_Solution_Math_Chapter_7_final_html_m1930c56c.gif)
Hence, the correct answer is D.
Page No 354:
Question 44:
The value of is
A. 1
B. 0
C. − 1
![](https://ddtarget.com/wp-content/uploads/2024/01/image-76.png)
ANSWER:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7935/NCERT_Solution_Math_Chapter_7_final_html_m470456ac.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7935/NCERT_Solution_Math_Chapter_7_final_html_1361c568.gif)
Adding (1) and (2), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/236/7935/NCERT_Solution_Math_Chapter_7_final_html_m7743de6a.gif)
Hence, the correct answer is B.