Question

Solve : $\int \frac{\log \left(\tan \frac{x}{2}\right)}{\sin x}dx$

Answer 1

$=\int \frac{\log \left(\tan \frac{x}{2}\right)}{\sin x}dx$

$\log \left(\tan \frac{x}{2}\right)=t$

differentiating

$\left[\frac{1}{\tan \frac{x}{2}}\right]\times {\sec }^2\frac{x}{2}\times \frac{1}{2}dx=dt\Rightarrow \left(\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}\right)\times \frac{1}{2\times \left({\cos }^2\frac{x}{2}\right)}dx=dt\Rightarrow \frac{1}{\sin x}dx=dt$

$I=\int \frac{\log \left(\tan \frac{x}{2}\right)}{\sin x}dx=\int tdt\Rightarrow \frac{t^2}{2}+c$

$I=\frac{1}{2}{\left[\log \left(\tan \frac{x}{2}\right)\right]}^2+c$