Question

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

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

Answer 1

Answer: (C)

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

Answer 2

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