Question

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

Answer 1

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

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

$=\int e^{x}2\tan xdx+$$2\int e^x{\sec }^{2}xdx$

$=2e^x\tan x-2\int e^x{\sec }^{2}xdx+$$2\int e^x{\sec }^{2}xdx+c$ [$c$ being integrating constant] [Using integration by parts]

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