Mathematics

The value of $$\int { \sin { x } \log { \left( \cot { \dfrac { x }{ 2 }  }  \right)  }  } dx$$ is


SOLUTION
Put, $$\log(\cot \dfrac{x}{2}) = t $$

$$\implies \dfrac{1}{2}\dfrac{1}{\cot \dfrac{x}{2}} (- \csc^{2} \dfrac{x}{2}) dx = dt$$

$$\implies \dfrac{-1}{2} (\tan \dfrac{x}{2} )\left(\dfrac{1}{\sin^{2} \dfrac{x}{2}}\right) dx = dt$$

$$\implies \dfrac{- 1}{2\sin \dfrac{x}{2} \cos \dfrac{x}{2}} dx = dt$$

$$\implies \dfrac{-1}{\sin x} dx = dt$$

Now, $$\int \dfrac{\log \left(\cot \dfrac{x}{2}\right)}{\sin x} dx = \int - t dt$$

$$ = - \dfrac{t^2}{2} + c$$

$$ = - \dfrac{1}{2} \log \left(\cot \dfrac{x}{2}\right) + c$$
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Subjective Medium Published on 17th 09, 2020
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