Evaluate the following integrals: ^3x dx - Vidyadip

Question

Evaluate the following integrals:

$\int\text{cosec}^3\text{x dx}$

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Answer

Let $\text{I}=\int\text{cosec}^3\text{dx}$
$=\int\text{cosec x}-\text{cosec}^2\text{x dx}$
using integration by parts,
$=\text{cosec x}\times\int\text{cosec}^2\text{x dx}+\int(\text{cosec x}\cot\text{x}\int\text{cosec}^2\text{x dx})\text{dx}$
$=\text{cosec x}\times(-\cot\text{x})+\int\text{cosec x}\cot\text{x}(-\cot\text{x})\text{dx}$
$=-\text{cosec x}\cot\text{x}-\int\text{cosec x}\cot^2\text{x dx}$
$=-\text{cosec x}\cot \text{x}-\int\text{cosec x}(\text{cosec}^2\text{x}-1)\text{dx}$
$=-\text{cosec x}\cot\text{x}-\int\text{cosec}^3\text{x dx}+\int\text{cosec x dx}$
$\text{I}=-\text{cosec x}\cot\text{x}-\text{I}+\log\Big|\tan\frac{\text{x}}{2}\Big|+\text{C}_1$
$2\text{I}=-\text{cosec x}\cot\text{x}+\log\Big|\tan\frac{\text{x}}{2}\Big|+\text{C}_1$
$\text{I}=-\frac{1}{2}\text{cosec x}\cot\text{x}+\frac{1}{2}\log\Big|\tan\frac{\text{x}}{2}\Big|+\text{C}$

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