Evaluate: x [3] x d x - Vidyadip

Question

Evaluate: $\int \frac{\cot x}{\sqrt[3]{\sin x}} d x$

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Answer

$\text { (a) : Let } I=\int \frac{\cot x}{\sqrt[3]{\sin x}} d x=\int \frac{\cos x}{\sin ^{1 / 3} x \cdot \sin x} d x$
$=\int \frac{\cos x}{\sin ^{4 / 3} x} d x=\int \sin ^{-4 / 3} x \cdot \cos x d x$
Put $\sin x=t \Rightarrow \cos x d x=d t$
$\Rightarrow I=\int t^{-4 / 3} d t=\frac{t^{-1 / 3}}{-1 / 3}+C=\frac{-3}{\sqrt[3]{\sin x}}+C$

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