`int csc^4(x) cot^6(x) dx` Evaluate the integral

`intcsc^4(x)cot^6(x)dx`

Rewrite the integrand by applying the identity,`1+cot^2(x)=csc^2(x)`

`=int(1+cot^2(x))^2cot^6(x)dx`

Apply integral substitution,

Let `u=cot(x)`

`=>du=-csc^2(x)dx`

`=>dx=(-1/(csc^2(x)))du=-1/(1+u^2)du`

`=int(1+u^2)^2u^6(-1/(1+u^2))du`

`=-int(1+u^2)u^6du`

`=-int(u^6+u^8)du`

`=-(intu^6du+intu^8du)`

`=-(u^7/7+u^9/9)`

`=-1/7u^7-1/9u^9`

Substitute back `u=cot(x)`

`=-1/7cot^7(x)-1/9cot^9(x)`

Add a constant C to the solution,

`=-1/7cot^7(x)-1/9cot^9(x)+C`