Question

Evaluate: $\int e^x\left(\frac{1-\sin x}{1-\cos x}\right)dx$

Answer 1

Consider the given integral.

$I=\int e^x\frac{1-\sin x}{1-\cos x}dx$

$I=\int \frac{e^x(1-2\sin \frac{x}{2}\cos \frac{x}{2})}{2{\sin }^2\frac{x}{2}}dx$

$I=\int [\frac{e^x}{2}\csc \frac{x}{2}-e^x\cot \frac{x}{2}]dx$

$I=\int \frac{e^x}{2}\csc \frac{x}{2}dx-\int e^x\cot \frac{x}{2}dx$

$I=\int \frac{e^x}{2}\csc \frac{x}{2}dx-\int e^x\cot \frac{x}{2}dx$

$I=\int \frac{e^x}{2}\csc \frac{x}{2}dx-\int \frac{e^x}{2}\csc \frac{x}{2}dx-e^x\cot \frac{x}{2}$

$I=-e^x\cot \frac{x}{2}+C$

Hence, this is the answer.