Question

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

Answer 1

Our required integral is,

`I = intcot^5xcsc^2xdx`

First, we make the substitution,

`cotx=t`

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

`d/dt(cotx)=dt/dt`

Then we get,

`(-csc^2x)dx/dt=1`

On rearraging,

`csc^2xdx = -dt`

On substituting these values in our original integral, we get

`I = intt^5dt`

which can be solved trivially to get the solution

`t^6/6 + C`

On substituting the value of `t = cotx`, we get,

`I = cot^6x/6+C`

In case your question was `intcotx^5cscx^2dx` the answer is far more complicated and beyond the scope of a straight-forward pen and paper solution. Maybe, the solution does not even exist.