Question

Integrate $\frac{e^x.\log \left(\sin e^x\right)}{\tan \left(e^x\right)}dx$

Answer 1

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

Solving by substitution,

Put, $e^x=t$

therefore, $e^x.dx=dt$

=> $\int \cot \left(t\right).\log \left(\sin t\right)dt$

=> $\int \frac{\cos \left(t\right).\log \left(\sin t\right)}{\sin t}dt$

Put , $\sin t=p$

therefore, $\cos tdt=dp$

=> $\int \frac{\log \left(p\right)}{p}dp$

=> Again substituting, $\log \left(p\right)=s$

=> $\frac{1}{p}dp=ds$

=>$\int s.ds$

=>$\frac{s^2}{2}+C$

On putting values of all taken variables , We get

=> $\frac{{\log }^2\left(\sin e^x\right)}{2}+C$