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AP EAMCET201921 Apr 2019Evening ShiftMathematicsIndefinite IntegrationActual

For n ≥ 2 , if I n = ∫ ( sin x + cos x ) n d x , then n I n - 2 ( n - 1 ) I n - 2 =

Options

  1. A( sin x + cos x ) n + 1 ( sin x - cos x ) + C
  2. B( sin x + cos x ) n ( sin x - cos x ) + C
  3. C( sin x + cos x ) n - 1 ( sin x - cos x ) + C
  4. D( sin x - cos x ) n - 1 ( sin x + cos x ) + C

Correct answer

C. ( sin x + cos x ) n - 1 ( sin x - cos x ) + C

Step-by-step solution

The integral expression is given as, I n = ∫ ( sin x + cos x ) n d x ⇒ I n = ∫ ( sin x + cos x ) n - 1 · ( sin x + cos x ) d x As we know from integration by parts ∫ u v d x = u ∫ v d x - ∫ d u d x · ∫ v d x d x ⇒ I n = ( sin x + cos x ) n - 1   ( sin x - cos x ) - ( n - 1 ) ∫ ( sin x + cos x ) n - 2 cos x - sin x · ( sin x - cos x ) d x ∫ sin x d x = - cos x ,   ∫ cos x d x = sin x ⇒ I n = ( sin x + cos x ) n - 1 ( sin

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