Question

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

• A. $e^x\sin x.$
• B. $e^x\cos x.$
• C. $e^x\tan x.$
• D. $e^x\cos 2x.$

Answer 1

Answer: (C)

$e^x\tan x.$

$I=\int e^x\frac{(2+\sin 2x)}{(1+\cos 2x)}dx.$
$=\int [\frac{e^x.2}{2{\cos }^{2}x}+e^x\frac{2\sin x\cos x}{2{\cos }^{2}x}]dx$
$\int e^x{\sec }^{2}xdx+\int e^x\tan xdx.$
Integrate first by parts
$I=e^x\tan x-\int e^x\tan xdx+\int e^x\tan xdx.$
$=e^x\tan x.$ The last two integrals cancel.
Above is of the form
$\int e^x[f\left(x\right)+f^{\prime }\left(x\right)]dx=e^{x}f\left(x\right),$
where $f\left(x\right)=\tan x$ and $f\left(x\right)={\sec }^{2}x.$