AP EAMCET · Maths · Indefinite Integration
\(\int e^{\sin x} \frac{\left(x \cos ^3 x-\sin x\right)}{\cos ^2 x} d x=\)
- A \(e^{\sin x}(x-\sec x)+C\)
- B \(e^{\sin x}(x-\operatorname{cosec} x)+C\)
- C \(e^{\sin x}(x+\sec x)+C\)
- D \(e^{\sin x}(x+\operatorname{cosec} x)+C\)
Answer & Solution
Correct Answer
(A) \(e^{\sin x}(x-\sec x)+C\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text {} \quad I=\int e^{\sin x}\left(\frac{x \cos ^3 x-\sin x}{\cos ^2 x}\right) d x \\ & =\int e^{\sin x}(x \cos x-\tan x \sec x) d x \\ & =\int e^{\sin x} x \cdot \cos x d x-\int e^{\sin x} \cdot \sec x \tan x d x \\ & =x \int e^{\sin x} \cos x d…
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