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Q. $\int\left(\frac{2-\sin 2 x}{1-\cos 2 x}\right) e^{x} d x$ is equal to

ManipalManipal 2009

Solution:

Let $I=\int\left(\frac{2-\sin 2 x}{1-\cos 2 x}\right) e^{x} \,d x$
$=\int\left(\frac{2-2 \sin x \cos x}{2 \sin ^{2} x}\right) e^{x} \,d x$
$=\int \underset{II}{\text{cosec}^{2}} \,\underset{I}{x \,e^{x}\, d x}-\int \cot\, x \,e^{x} \,d x$
$=-\cot \,x \,e^{x}-\int(-\cot x) e^{x} d x-\int \cot\, x e^{x} \,d x+c$
$=-\cot \,x \,e^{x}+c$