Question

Solve: $\int \log \left(\tan \left(\frac{x}{2}\right)\right)\csc xdx$.

Answer 1

$I=\int log\left(\tan \frac{x}{2}\right).cosecxdx$

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

$\Rightarrow \frac{1}{\tan \frac{x}{2}}.{\sec }^2\frac{x}{2}.\frac{1}{2}dx=dt$

$\Rightarrow \frac{\left(cos\frac{x}{2}\right)dx}{2\sin \frac{x}{2}.{\cos }^2\frac{x}{2}}=dt$

$\Rightarrow \frac{dx}{sinx}=dt$

$I=\int tdt=\frac{t^2}{2}+C$

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