Question

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

Answer 1

$\begin{array}{cc}\int e^x\left(\frac{1+\sin x}{1+\cos x}\right)dx& \\ \Rightarrow \int \left[\frac{1}{1+\cos x}+\frac{\sin x}{1+\cos x}\right]e^{x}dx& \\ \Rightarrow \int \left[\frac{1}{2{\cos }^2\frac{x}{2}}+\frac{2\sin \frac{x}{2}.\cos \frac{x}{2}}{2{\cos }^2\frac{x}{2}}\right]e^{x}dx& \\ \Rightarrow \int \left[\frac{{\sec }^2\frac{x}{2}}{2}+\tan \frac{x}{2}\right]e^{x}dx& \\ \Rightarrow \int \left[f^{\prime }\left(x\right)+f\left(x\right)\right]e^{x}dx& \left[\int e^x\left[f\left(x\right)+f^{\prime }\left(x\right)\right]dx=e^{x}f\left(x\right)+c\right]\\ \Rightarrow e^x.f\left(x\right)+c& \\ e^x.\tan \frac{x}{2}+c.& \end{array}$

Hence, this is the answer.