Question #30ad4

1 Answer
May 15, 2017

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