Question

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

Answer 1

Consider, $I=\int e^x\left(\frac{1+\sin x}{1+\cos x}\right)dx$

$=\int e^x\left(\frac{1+2\sin \frac{x}{2}.\cos \frac{x}{2}}{2{\cos }^2\frac{x}{2}}\right)dx$

$=\int e^x(\frac{1}{2}{\sec }^2\frac{x}{2}+\tan \frac{x}{2})dx$

$=\int e^x\tan \frac{x}{2}dx$$+\int \frac{e^x}{2}{\sec }^2\frac{x}{2}dx$

$=e^x\tan \frac{x}{2}$$-\int \frac{e^x}{2}{\sec }^2\frac{x}{2}dx$$+\int \frac{e^x}{2}{\sec }^2\frac{x}{2}dx+c$ [$c$ being integrating constant] [Using by-parts method]

$=e^x\tan \frac{x}{2}+c$.