Question

Integrate the function $\frac{2+\sin 2x}{1+\cos 2x}{e}^x$

Answer 1

$I=\int \left(\frac{2+\sin 2x}{1+\cos 2x}\right)e^x$
$=\int \left(\frac{2+2\sin x\cos x}{2{\cos }^{2}x}\right)e^x$
$=\int \left(\frac{1+\sin x\cos x}{{\cos }^{2}x}\right)e^x$
$=\int ({\sec }^{2}x+\tan x)e^x$
Let $f\left(x\right)=\tan x\Rightarrow f^{\prime }\left(x\right)={\sec }^{2}x$
$\therefore I=\int \left(f\right(x)+f^{\prime }(x\left)\right]e^{x}dx$
$=e^{x}f\left(x\right)+C$
$=e^x\tan x+C$