January 22, 2024

Unlock the power of NCERT Solutions for Class 12 Science Maths Chapter 1: Integrals! Dive into step-by-step explanations that make tackling homework and exam prep a breeze. Access all the questions and answers from the NCERT Book for Class 12 Science Maths Chapter 1, absolutely free by DD Target PMT 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, 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, 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, Therefore, the anti derivative of . Page No 299: Question 4: ANSWER: The anti derivative of is the function of x whose derivative is . It is known that, Therefore, the anti derivative of . Page No 299: Question 5: ANSWER: The anti derivative of  is the function of x whose derivative is . It is known that, Therefore, the anti derivative of  is . Page No 299: Question 6: ANSWER: Page No 299: Question 7: ANSWER: Page No 299: Question 8: ANSWER: Page No 299: Question 9: ANSWER: Page No 299: Question 10: ANSWER: Page No 299: Question 11: ANSWER: Page No 299: Question 12: ANSWER: Page No 299: Question 13: ANSWER: On dividing, we obtain Page No 299: Question 14: ANSWER: Page No 299: Question 15: ANSWER: Page No 299: Question 16: ANSWER: Page No 299: Question 17: ANSWER: Page No 299: Question 18: ANSWER: Page No 299: Question 19: ANSWER: Page No 299: Question 20: ANSWER: Page No 299: Question 21: The anti derivative of equals ANSWER: 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, ∴Anti derivative of  ∴ Also, Hence, the correct answer is A. Page No 304: Question 1: ANSWER: Let = t ∴2x dx = dt Page No 304: Question 2: ANSWER: Let log |x| = t ∴  Page No 304: Question 3: ANSWER: Let 1 + log x = t ∴  Page No 304: Question 4: sin x ⋅ sin (cos x) ANSWER: sin x ⋅ sin (cos x) Let cos x = t ∴ −sin x dx = dt Page No 304: Question 5: ANSWER: Let  ∴ 2adx = dt Page No 304: Question 6: ANSWER: Let ax + b = t ⇒ adx = dt Page No 304: Question 7: ANSWER: Let  ∴ dx = dt Page No 304: Question 8: ANSWER: Let 1 + 2×2 = t ∴ 4xdx = dt Page No 304: Question 9: ANSWER: Let  ∴ (2x + 1)dx = dt Page No 304: Question 10: ANSWER: Let  ∴ Page No 304: Question 11: ANSWER: Page No 304: Question 12: ANSWER: Let  ∴  Page No 304: Question 13: ANSWER: Let  ∴ 9x2dx = dt Page No 304: Question 14: ANSWER: Let log x = t ∴  Page No 304: Question 15: ANSWER: Let  ∴ −8x dx = dt Page No 304: Question 16: ANSWER: Let  ∴ 2dx = dt Page No 304: Question 17: ANSWER: Let  ∴ 2xdx = dt Page No 305: Question 18: ANSWER: Let  ∴  Page No 305: Question 19: ANSWER: Dividing numerator and denominator by ex, we obtain Let  ∴  Page No 305: Question 20: ANSWER: Let  ∴  Page No 305: Question 21: ANSWER: Let 2x − 3 = t ∴ 2dx = dt Page No 305: Question 22: ANSWER: Let 7 − 4x = t ∴ −4dx = dt Page No 305: Question 23: ANSWER: Let  ∴  Page No 305: Question 24: ANSWER: Let  ∴  Page No 305: Question 25: ANSWER: Let  ∴  Page No 305: Question 26: ANSWER: Let  ∴  Page No 305: Question 27: ANSWER: Let sin 2x = t ∴  Page No 305: Question 28: ANSWER: Let  ∴ cos x dx = dt Page No 305: Question 29: cot x log sin x ANSWER: Let log sin x = t Page No 305: Question 30: ANSWER: Let 1 + cos x = t ∴ −sin x dx = dt Page No 305: Question 31: ANSWER: Let 1 + cos x = t ∴ −sin x dx = dt Page No 305: Question 32: ANSWER: Let sin x + cos x = t ⇒ (cos x − sin x) dx = dt Page No 305: Question 33: ANSWER: Put cos x − sin x = t ⇒ (−sin x − cos x) dx = dt Page No 305: Question 34: ANSWER: Page No 305: Question 35: ANSWER: Let 1 + log x = t ∴  Page No 305: Question 36: ANSWER: Let  ∴  Page No 305: Question 37: ANSWER: Let x4 = t ∴ 4×3 dx = dt Let  ∴ From (1), we obtain Page No 305: Question 38: equals ANSWER: Let  ∴  Hence, the correct answer is D. Page No 305: Question 39: equals A.  B.  C.  ANSWER: Hence, the correct answer is B. Page No 307: Question 1: ANSWER: Page No 307: Question 2: ANSWER: It is known that,  Page No 307: Question 3: cos 2x cos 4x cos 6x ANSWER: It is known that, Page No 307: Question 4: sin3 (2x + 1) ANSWER: Let  Page No 307: Question 5: sin3x cos3x ANSWER: Page No 307: Question 6: sin x sin 2x sin 3x ANSWER: It is known that,  Page No 307: Question 7: sin 4x sin 8x ANSWER: Page No 307: Question 8: ANSWER: Page No 307: Question 9: ANSWER: Page No 307: Question 10: sin4x ANSWER: Page No 307: Question 11: cos4 2x ANSWER: Page No 307: Question 12: ANSWER: Page No 307: Question 13: ANSWER: Page No 307: Question 14: ANSWER: Page No 307: Question 15: ANSWER: Page No 307: Question 16: tan4x ANSWER: From equation (1), we obtain Page No 307: Question 17: ANSWER: Page No 307: Question 18: ANSWER: Page No 307: Question 19: ANSWER: Page No 307: Question 20: ANSWER: Page No 307: Question 21: sin−1 (cos x) ANSWER: It is known that, Substituting in equation (1), we obtain Page No 307: Question 22: ANSWER: 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: 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: Let exx = t Hence, the correct answer is B. Page No 315: Question 1: ANSWER: Let x3 = t ∴ 3x2dx = dt …

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