Evaluate the following integrals: x ( cosec x- ) dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{cosec x}\log({\text{cosec x}-\cot\text{x})}\text{dx}$

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Answer

Let $\text{I}=\int\text{cosec x}\log(\text{cosec x}-\cot\text{x})\text{dx}\ ....(1)$
Let $\log(\text{cosec x}-\cot\text{x})=\text{t}$ then,
$\text{dx}[\log(\text{cosec x}-\cot\text{x})]=\text{dt}$
$\Rightarrow\text{cosec x}\text{ dx}=\text{dt}$ $\Big[\because\ \frac{\text{d}}{\text{dx}}(\log(\text{cosec x}-\cot\text{x}))=\text{cosec x}\Big]$
Putting $\log(\text{cosec x}-\cot\text{x})=\text{t}$ and $\text{cosec x}\text{ dx}=\text{dt}$ in equation (1), we get
$\text{I}=\int\text{t dt}$
$=\frac{\text{t}^2}{2}+\text{C}$
$\text{I}=\frac{1}{2}[\log(\text{cosec x}-\cot\text{x})]^2+\text{C}$

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