Question

Evaluate: $\int e^x\frac{(2+\sin 2x)}{{\cos }^{2}x}dx$

Answer 1

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

$=\int e^x(2{\sec }^{2}x+2\tan x)dx$

$=2\int e^x({\sec }^{2}x+\tan x)dx$

$=2\left[\int e^x{\sec }^{2}xdx+\int e^x\tan xdx\right]$

$=2\left[e^x\int {\sec }^{2}xdx-\int \left\{\frac{d}{dx}{e}^x\int {\sec }^{2}xdx\right\}dx+\int e^x\tan xdx\right]$

$=2[e^x\tan x-\int e^x\tan xdx+\int e^x\tan xdx]$

$=2e^x\tan x+c$