Question

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

Answer 1

$\int e^x\left[\frac{\cos x-\sin x}{{\sin }^{2}x}\right]dx=\int e^x(\frac{\cos x}{\sin x\times \sin x}-\frac{1}{\sin x})dx=-\int e^x(\text{cosec}x-\text{cosec}x\cot x)dx$

This is in the form of $\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)=e^{x}f\left(x\right)+C$

where $f\left(x\right)=\text{cosec}x$

$⟹-e^x\text{cosec}x+C$