Question

Evaluate the integral:

$\int e^x\left(\frac{1+sinx}{1+cosx}\right)dx$

Answer 1

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

It is of the form $\int e^x\left(f\right(x)+f^{\prime }(x\left)dx\right)=e^{x}f\left(x\right)+c$

Putting, $f\left(x\right)=\tan \frac{x}{2}$
$\therefore f^{\prime }\left(x\right)={\sec }^2\frac{x}{2}\left(\frac{1}{2}\right)=\frac{{\sec }^2\frac{x}{2}}{2}$
Thus,
$\int e^x(\frac{{\sec }^2\frac{x}{2}}{2}+\tan \frac{x}{2})dx=\int e^x\left(\frac{1+\sin x}{1+\cos x}\right)dx=e^x\tan \frac{x}{2}$