Question

Evaluate: $\int e^x(\tan x+\log (\sec x\left)\right)dx.$

Answer 1

$\int e^x(\tan x+\log \sec x)dx=\int e^x\tan xdx+\int e^x\log \sec xdx$
$=\int e^x\tan xdx+\log \left(\sec x\right)e^x-\int \frac{d}{dx}\left(\log \sec x\right).\int e^{x}dx$
$=\int e^x\tan xdx+\log \left(\sec \right)x{e}^x-\int e^x\tan xdx$
$=e^x\log \sec x+c$