Question

Evaluate $\int e^x\left(\tan x-\log \cos x\right)dx$

Answer 1

It is just an observation to make perfect differential;

it is in the form of $\int \left(e^x\right(f\left(x\right))+e^x(\frac{d}{dx}f\left(x\right)\left)\right)dx=e^x\left(f\right(x\left)\right)$

$\frac{d}{dx}\log \left(\cos x\right)=-\tan x$

so here $f\left(x\right)=\log \left(\cos x\right)$

so,

$\int e^x(\tan x-\log \cos x)dx=e^x(-\log \cos x)$