Integrate the function: cot x log sin x - Vidyadip

Question

Integrate the function: cot x log sin x

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Answer

Let $I = \int {\cot x\log \sin xdx} $ ...(i)

Putting log sin x = t

$ \Rightarrow \frac{1}{{\sin x}}\frac{d}{{dx}}\left( {\sin x} \right) = \frac{{dt}}{{dx}}$

$ \Rightarrow \frac{1}{{\sin x}}\cos x = \frac{{dt}}{{dx}}$

$ \Rightarrow \cot xdx = dt$

$\therefore$ From eq. (i), $I = \int {tdt} $

$ = \frac{{{t^2}}}{2} + c$

$= \frac{1}{2}{\left( {\log \sin x} \right)^2} + c$

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