Question

The value of $\int e^x\left[\frac{1+\sin x}{1+\cos x}\right]dx$ is equal to

• A. $e^x\tan \frac{x}{2}+c$
• B. $e^x\tan x+c$
• C. $e^x(1+\cos x)+c$
• D. $e^x(1+\sin x)+c$

Answer 1

Answer: (A)

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