# Evaluate the following integral. \int \frac{1}{6} csc (2x) cot(2x) \space dx

## Question:

Evaluate the following integral.

{eq}\int \frac{1}{6} csc (2x) cot(2x) \space dx {/eq}

## Evaluating Integral:

In this question, we have to evaluate the integral. And to evaluate, we use trigonometric substitution, and finally we do back substitution.

## Answer and Explanation:

{eq}\int \frac{1}{6}\csc \left(2x\right)\cot \left(2x\right)dx {/eq}

{eq}=\frac{1}{6}\cdot \int \:\csc \left(2x\right)\cot \left(2x\right)dx {/eq}

{eq}\mathrm{Apply\:u-substitution:}\:u=\csc \left(2x\right) \\ \frac{du}{dx}=-2\cot \left(2x\right)\csc \left(2x\right) {/eq}

So we will get

{eq}=\frac{1}{6}\cdot \int \:-\frac{1}{2}du \\ =\frac{1}{6}\left(-\frac{1}{2}\right)u+C= \frac{1}{6}\left(-\frac{1}{2}\right)\csc \left(2x\right)+C \\ =-\frac{1}{12}\csc \left(2x\right)+C {/eq}

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