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99 Percentile Qs Bank for JEE MainMathematicsIndefinite Integration

If ∫ 3 sin x + 2 cos x 3 cos x + 2 sin x dx = ax + b  ln  2 sin x + 3 cos x + C , then

Options

  1. Aa = - 12 13 , b = 15 39
  2. Ba = - 7 13 , b = 6 13
  3. Ca = 12 13 , b = - 15 39
  4. Da = - 7 13 , b = - 6 13

Correct answer

C. a = 12 13 , b = - 15 39

Step-by-step solution

Differentiating both sides, we get 3 sin x + 2 cos x 3 cos x + 2 sin x = = a + b(2 cos x - 3 sin x) (2 sin x + 3 cos x) = sin x (2a - 3b) + cos x (3a + 2b) 3 cos x + 2 sin x Comparing like terms on both sides, we get 3 = 2 a - 3 b, 2 = 3a + 2b ⇒ a = 1 2 1 3 , b = - 1 5 3 9

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