Question

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

Answer 1

$sinx=2sin\left(\frac{x}{2}\right)cos\left(\frac{x}{2}\right)$

$1+cosx=2co{s}^2\left(\frac{x}{2}\right)$

$\int e^x\left(tan\right(\frac{x}{2})+\frac{1}{2}se{c}^2(\frac{x}{2}\left)\right)dx$

$\int e^{x}tan\frac{x}{2}dx+\frac{1}{2}\int e^{x}se{c}^2\frac{x}{2}dx$

Integrating by parts

$u=e^x,v=se{c}^2\frac{x}{2}$

$u^{{}^{\prime }}=e^x,\int vdx=2tan\frac{x}{2}$

$\int e^{x}tan\frac{x}{2}dx+\frac{1}{2}{e}^{x}2tan\frac{x}{2}-\frac{1}{2}\int e^{x}2tan\frac{x}{2}dx$

$⟹e^{x}tan\frac{x}{2}$