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

Question

Evaluate the following integrals:
$\int\frac{\sin^3\text{x}-\cos^3\text{x}}{\sin^2\text{x}\cos^2\text{x}}\text{dx}$

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Answer

$\int\frac{\sin^3\text{x}-\cos^3\text{x}}{\sin^2\text{x}\cos^2\text{x}}\text{dx}$
$=\int\bigg(\frac{\sin^3\text{x}}{\sin\text{x}^2\cos^2\text{x}}-\frac{\cos^3\text{x}}{\sin\text{x}^2\cos^2\text{x}}\bigg)\text{dx}$
$=\int(\sin\text{x}\sec^2\text{x}-\cos\text{x }\text{cosec}^2\text{x})\text{dx}$
$=\int(\tan\text{x}\sec\text{x}-\cot\text{x }\text{cosec x})\text{dx}$
$=\sec\text{x}+\text{cosec x}+\text{C}$

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