# Evaluate the integral of cotx5 cscx2

We have to find the integral of $\int \cot x^{5}. \csc x^{2}$

### Solution

$\int \cot x^{5}. \csc x^{2}$

Let us now substitute

cot x = t

If we differentiate both sides with respect to t we get,

$\frac{\mathrm{d} }{\mathrm{d} x} cot x=\frac{dt}{dt}$

Then we get,

$-\csc^{2}x\frac{dx}{dt}= 1$

On rearraging,

$\csc^{2}x dx = -dt$

On substituting these values in the given integral, we get

$I=\int t^{5}dt$ $I=\int t^{5}dt= \frac{t^{6}}{6} + C$

On substituting the value of t = cot x we get

$I=\frac{cot^{6}x}{6} + C$ $\int \cot x^{5}. \csc x^{2}= \frac{cot^{6}x}{6} + C$

$\int \cot x^{5}. \csc x^{2}= \frac{cot^{6}x}{6} + C$