Question

Evaluate : $\int e^x\frac{\log \left(\sin e^x\right)}{\tan e^x}dx$

Answer 1

$I=\int e^x\frac{\log \left(\sin e^x\right)}{\tan e^x}dx$

Let $\log \left(\sin e^x\right)=t$

$\therefore e^x\frac{\cos x}{\sin x}dx=dt\Rightarrow e^x\cot xdx=dt$

Now, $I=\int e^x\frac{\log \left(\sin e^x\right)}{\tan e^x}dx=\int \log \left(\sin e^x\right)\left[e^x\cot e^x\right]dx=\int tdt$

$=\frac{t^2}{2}+c$
$=\frac{1}{2}\left[\log \right(\sin e^x){]}^2+c$
Where $c$ is a constant of Integration.