Evaluate: e^x [ x- x ^2 x ] d x - Vidyadip

Question

Evaluate: $\int e^x\left[\frac{\cos x-\sin x}{\sin ^2 x}\right] d x$

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Answer

$\begin{aligned} & I=\int e^x\left[\frac{\cos x}{\sin ^2 x}-\frac{\sin x}{\sin ^2 x}\right] d x \\ & =\int e^x\left[\begin{array}{cc}\cot x \cdot \cos e c x & -\cos e c x \\ f \prime(x) & f(x)\end{array}\right] \\ & \because \int e^x[f(x)+f \prime(x)] d x=e^x f(x)+C \\ & \therefore I=-e^x \cdot \cos e c x+C\end{aligned}$

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