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Q.
The value of $\displaystyle \int \frac{ln \left(c o t x\right)}{sin 2 x}dx$ is equal to (where, $C$ is the constant of integration)
NTA AbhyasNTA Abhyas 2020Integrals
Solution:
$\displaystyle \int \frac{ln \left(cot x\right)}{2 sin x cos x} d x = Ι$ (let)
Put $ln \left(cot x\right)=t$
$\Rightarrow \frac{1}{cot x}\times cosec^{2}xdx=-dt$
$\Rightarrow \frac{d x}{cos x sin x}=-dt$
So, $Ι=\frac{1}{2}\displaystyle \int \left(- t\right) d t=\frac{- 1}{2}\left(\frac{t^{2}}{2}\right)+C$
$=-\frac{1}{4}\left(ln \left(cot x\right)\right)^{2}+C$