Evaluate the following integrals: (a +b x)^2dx - Vidyadip

Question

Evaluate the following integrals:
$\int(\text{a}\tan\text{x}+\text{b}\cot \text{x})^2\text{dx}$

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Answer

$\int(\text{a}\tan\text{x}+\text{b}\cot \text{x})^2\text{dx}$
$=\int(\text{a}^2\tan^2\text{x}+\text{b}^2\cot^2\text{x}+2\text{ab}\tan\text{x}\cot\text{x})\text{dx}$
$=\text{a}^2\int\tan^2\text{x dx}+\text{b}^2\int\cot^2\text{x dx}+2\text{ab}\int\text{dx}$
$=\text{a}^2\int(\sec^2\text{x}-1)\text{dx}+\text{b}^2\int(\text{cosec}^2\text{x}-1)\text{dx}+2\text{ab}\int\text{dx}$
$=\text{a}^2[\tan\text{x}-\text{x}]+\text{b}^2[-\cot\text{x}-\text{x}]+2\text{ab}\text{x}+\text{C}$
$=\text{a}^2\tan\text{x}-\text{b}^2\cot\text{x}-(\text{a}^2+\text{b}^2-2\text{ab})\text{x}+\text{C}$

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