Question

Question #30ad4

Answer 1

`int tan^2(x)csc^3(x)=csc(x) -ln abs(csc(x)+cot(x))+C`

Explanation:

`int tan^2(x)csc(x)^3 dx=int (sin^2(x)/cos^2(x))(1/sin^3(x)) dx=int 1/(cos^2(x)sin(x)) dx = int (sin^2(x)+cos^2(x))/(cos^2(x)sin(x)) dx =int sin^2(x)/(cos^2(x)sin(x)) dx + int cos^2(x)/(cos^2(x)sin(x)) dx`

`( 1st )` integral:
`int sin^2(x)/(cos^2(x)sin(x)) dx =int sin(x)/cos^2(x) dx`

Substitute `u=cosx`
also `d/dxcos(x)=-sin(x)`
then by Substitution Rule:

`-int1/u^2 du=-(-2+1)(u)^(-2+1)=u^-1=1/cos(x)=csc(x)`

`(2nd)` integral:
`int 1/sin(x) dx=int csc(x) dx = -ln abs(csc(x)+cot(x))`
This is a common integral.

Putting it all together:
`int tan^2(x)csc^3(x)=csc(x) -ln abs(csc(x)+cot(x))+C`