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AP EAMCET · Maths · Indefinite Integration

\(\int \frac{e^{\sin x}(\sin 2 x-8 \cos x)}{2(\sin x-3)^2} d x=\)

  1. A \(e^{\sin x}(\sin x-3)+c\)
  2. B \(\frac{e^{\sin x}}{(\sin x-3)^2}+c\)
  3. C \(e^{\sin x}(\sin x-3)^2+c\)
  4. D \(\frac{e^{\sin x}}{\sin x-3}+c\)
Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{e^{\sin x}}{\sin x-3}+c\)

Step-by-step Solution

Detailed explanation

\( \int \frac{e^{\sin x}(2 \sin x \cos x-8 \cos x)}{2(\sin x-3)^2} d x = \int e^{\sin x} \cos x \frac{\sin x-4}{(\sin x-3)^2} d x \) Let \( u = \sin x \). Then \( d u = \cos x d x \). The integral becomes \( \int e^u \frac{u-4}{(u-3)^2} d u \).…