Evaluate the following integrals: cosec^2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{x}\text{ cosec}^2\text{x dx}$

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Answer

Let $\text{I}=\int\text{x cosec}^2\text{x dx}$
Using integration by parts,
$\text{I}=\text{x}\int\text{cosec}^2\text{x dx}-\int(\int\text{cosec}^2\text{x dx})\text{dx}$
$=-\text{x}\cot\text{x}+\int\cot\text{x dx}$
$=-\text{x}\cot\text{x}+\log|\sin\text{x}|+\text{C}$

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