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

Question

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

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Answer

$\int\frac{\text{cosec }\text{x}}{\text{cosec }\text{x}-\cot\text{x}}\text{dx}$
$=\int\frac{\text{cosec }\text{x}}{\text{cosec }\text{x}-\cot\text{x}}\times\frac{\text{cosec x}+\cot\text{x}}{\text{cosec x}+\cot\text{x}}\times\text{dx}$
$=\int\frac{\text{cosec x}(\text{cosec x}+\cot\text{x})}{\text{cosec}^2\text{x}-\cot^2\text{x}}\text{dx}$
$=\int(\text{cosec}^2\text{x}+\text{cosec x}\cot\text{x})\text{dx}$
$=\int\text{cosec}^2\text{x dx}+\int\text{cosec x dx}$
$=-\cot\text{x}-\text{cosec x}+\text{C}$
$\therefore\ \int\frac{\text{cosec }\text{x}}{\text{cosec }\text{x}-\cot\text{x}}\text{dx}=-\cot\text{x}-\text{cosec x}+\text{C}$

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