# How do you find the integral of #cotx^5cscx^2#?

Our required integral is,

First, we make the substitution,

Then we get,

On rearraging,

On substituting these values in our original integral, we get

which can be solved trivially to get the solution

By signing up, you agree to our Terms of Service and Privacy Policy

To integrate ( \cot(x)^5 \csc(x)^2 ), you can use the substitution method. Let ( u = \cot(x) ). Then, ( du = -\csc(x)^2 dx ).

So, ( dx = -\frac{du}{\csc(x)^2} = -du \sin(x) ), and ( \cot(x) = u ).

Substituting these into the integral:

( \int u^5 (-du \sin(x)) )

( = -\int u^5 du \sin(x) )

Now, integrate ( u^5 ) with respect to ( u ):

( = -\frac{1}{6}u^6 \sin(x) + C )

Replace ( u ) with ( \cot(x) ):

( = -\frac{1}{6}\cot(x)^6 \sin(x) + C )

So, the integral of ( \cot(x)^5 \csc(x)^2 ) is ( -\frac{1}{6}\cot(x)^6 \sin(x) + C ).

By signing up, you agree to our Terms of Service and Privacy Policy

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7