Evaluate the following integrals: ^3x cosec^2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\cot^3\text{x }\text{cosec}^2\text{x}\text{ dx}$

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Answer

$\int\cot^3\text{x}\text{ cosec}^2\text{x}\text{ dx}$
$\text{Let},\cot\text{x}=\text{t}$
$\Rightarrow-\text{cosec}^2\text{x}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\text{cosec}^2\text{x}\text{ dx}=-\text{dt}$
$\text{Now},\int\cot^3\text{x}\text{ cosec}^2\text{x}\text{ dx}$
$=\int\text{t}^3(-\text{dt})$
$=\frac{-\text{t}^4}{4}+\text{C}$
$=\frac{-\cot^4\text{x}}{4}+\text{C}$

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