Question

If \(\int e^{\sin x}\left[\frac{x \cos ^{3} x-\sin x}{\cos ^{2} x}\right] d x=e^{\sin x} \cdot f(x)+c\), where \(c\) is constant of integration, then \(f(x)=\)
(A) \(\sec x-x\)
(B) \(x-\sec x\)
(C) \(\tan x-x\)
(D) \(x-\tan x\)

Answer 1

Ans: (B)
Hint \(: e^{\sin x}\left(\frac{x \cos ^{3} x-\sin x}{\cos ^{2} x}\right)=e^{\sin x}(x \cos x-\sec x \tan x)\)
$$
\begin{aligned}
&=e^{\sin x}(x \cos x-1+1-\sec x \tan x)=e^{\sin x} \times \cos x(x-\sec x)+e^{\sin x}(1-\sec x \tan x) \\
&=d\left[e^{\sin x}(x-\sec x)\right]
\end{aligned}
$$

Answer 2

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Answer 3

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