Question

$\int \frac{e^x(1+\sin x)}{(1+\cos x)}$dx is equal to

• A. $\log \tan x$
• B. $e^x\tan \left(x/2\right)$
• C. $\sin \log x$
• D. $e^x\cos x$

Answer 1

The correct option is B $e^x\tan \left(x/2\right)$

Let
$I=\int \frac{e^x(1+\sin x)}{1+\cos x}dx=\int \frac{e^x(1+2\sin \frac{x}{2}\cos \frac{x}{2})}{2{\cos }^2\frac{x}{2}}dx=\int e^x(\left(\frac{1}{2{\cos }^2\frac{x}{2}}\right)+\left(\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}\right))dx=\int e^x(\frac{1}{2}{\sec }^2\frac{x}{2}+\tan \frac{x}{2})dx$
Using $\int e^x(f\left(x\right)+f^{\prime }\left(x\right))dx=e^{x}f\left(x\right)+c$
We get $I=e^x\tan \frac{x}{2}+c$