Question

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

• A. $e^x\tan \left(\frac{x}{2}\right)+C$
• B. $\tan \left(\frac{x}{2}\right)+C$
• C. $e^x+C$
• D. $e^x\sin x+C$

Answer 1

The correct option is A $e^x\tan \left(\frac{x}{2}\right)+C$

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

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

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

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