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)

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