`int_(pi/4)^(pi/2) cot^5(x) csc^3(x) dx` Evaluate the integral

`int _(pi/4)^(pi/2) cot^5(x)csc^3(x)dx`

To solve, apply the Pythagorean identity  `1+cot^2(x)=csc^2(x)`      to express the integral in the form `int u^n du.`      

`=int _(pi/4)^(pi/2) cot^2 (x) cot^3(x) csc^3(x)dx`

`=int_(pi/4)^(pi/2) (csc^2(x)-1) cot^3(x)csc^3(x)dx`

`=int_(pi/4)^(pi/2)( csc^5(x)cot^3(x)-csc^3(x)cot^3(x) )dx`

`=int_(pi/4)^(pi/2) ( csc^5(x) cot(x) cot^2(x) - csc^3(x) cot(x)cot^2(x) )dx`

`=int_(pi/4)^(pi/2) ( csc^5(x) cot(x) (csc^2(x)-1) - csc^3(x) cot(x) (csc^2(x) -1)) dx`

`=int_(pi/4)^(pi/2) (csc^7(x)cot(x)-csc^5(x)cot(x)-csc^5(x)cot(x) +csc^3(x)cot(x)) dx`

`=int_(pi/4)^(pi/2) (csc^7(x)cot(x) -2csc^5(x)cot(x) +csc^3(x)cotx) dx`

To take the integral, apply u-substitution method. So, let u be:

     u=csc(x) dx

Then, differentiate u.

     du = - csc(x) cot(x) dx

To be able to apply this, factor out -csc(x) cot(x).

`=int_(pi/4)^(pi/2) ( -csc^6(x) +2csc^4(x) -csc^2(x)) (-csc(x) cot(x) dx)` 

Then, determine the value of u when x=pi/2 and x=pi/4.

     `u=csc (x)`

     `u= csc(pi/2)=1`

     `u=csc(pi/4)=sqrt2`

Expressing the integral in terms of u, the integral becomes:

`=int _sqrt2^1 (-u^6 +2u^4-u^2) du`

`=( -u^7/7 +(2u^5)/5 -u^3/3 )_sqrt2^1 `

`=(-1/7+2/5-1/3) -(-(sqrt2)^7/7+(2(sqrt2)^5)/5-(sqrt2)^3/3)`

`=0.2201`

Therefore, `int _(pi/4)^(pi/2) cot^5(x) csc^3(x) dx=0.2201`  .