Evaluate the following : ( x+ x)^2 d x - Vidyadip

Question

Evaluate the following : $\int(\tan x+\cot x)^2 \cdot d x$

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Answer

$\int(\tan x+\cot x)^2 \cdot d x$
$
\begin{aligned}
& =\int\left(\tan ^2 x+2 \tan x \cdot \cot x+\cot ^2 x\right) \cdot d x \\
& =\int\left(\tan ^2 x+2+\cot ^2 x\right) \cdot d x \\
& =\int\left(\sec ^2 x-1+2+\operatorname{cosec}^2 x-1\right) \cdot d x \\
& =\int\left(\sec ^2 x+\operatorname{cosec}^2 x\right) \cdot d x \\
& =\int \sec ^2 x \cdot d x+\int \operatorname{cosec}^2 x \cdot d x \\
& =\tan x+(-\cot x)+c \\
& =\tan x-\cot x+c
\end{aligned}
$

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