Evaluate: x ( x 2 + x 2 )^3 d x

Question

Evaluate: $\int \frac{\cos x}{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^3} d x$

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Answer

$(c):$ We have,
$ \int \frac{\cos x}{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^3} d x=\int \frac{\cos ^2(x / 2)-\sin ^2(x / 2)}{\{\cos (x / 2)+\sin (x / 2)\}^3} d x$
$\text { Put } t=\cos \frac{x}{2}+\sin \frac{x}{2}$
$\Rightarrow 2 d t=\left[\cos \frac{x}{2}-\sin \frac{x}{2}\right] d x$
$\Rightarrow \int \frac{\cos (x / 2)-\sin (x / 2)}{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^2} d x=2 \int \frac{1}{t^2} d t$
$\quad=\frac{-2}{t}+C=\frac{-2}{\cos (x / 2)+\sin (x / 2)}+C$

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