Evaluate: 2x+2 ^2x ^2x dx - Vidyadip

Question

Evaluate:
$\int\frac{\cos2\text{x}+2\sin^2\text{x}}{\sin^2\text{x}}\text{dx}$

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Answer

$\int\Big(\frac{\cos2\text{x}+2\sin^2\text{x}}{\sin^2\text{x}}\Big)\text{dx}$
$=\int\Big(\frac{1-2\sin^2\text{x}+2\sin^2\text{x}}{\sin^2\text{x}}\Big)\text{dx}$ $[\because\cos2\text{x}=1-2\sin^2\text{x}]$
$=\int\text{cosec}^2\text{x}\text{ dx}$
$=-\cot\text{x}+\text{c}$

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