Integrating ∫cos(x)^2*tan(x)^3dx using u-substitution and integration by parts

[NWeid1]

Homework Statement

∫cos(x)^2*tan(x)^3dx

Homework Equations

The Attempt at a Solution

Were learning Integration by parts and u substitution but this one I can't figure out. I tried making it ∫cos(x)*(sin(x)^3)/(cos(x)^3)dx and then ∫tan(x)*sin(x)^2 but I don't know if I'm going in the right direction because I don't know how to solve from here.

 

[SammyS]

Homework Statement

∫cos(x)^2*tan(x)^3dx

Homework Equations

The Attempt at a Solution

Were learning Integration by parts and u substitution but this one I can't figure out. I tried making it ∫(cos(x))^2*(sin(x)^3)/(cos(x)^3)dx and then ∫tan(x)*sin(x)^2 but I don't know if I'm going in the right direction because I don't know how to solve from here.

That should be \displaystyle \int\frac{\cos^2(x)\sin^3(x)}{\cos^3(x)}\,dx

The integrand can be simplified to:

\displaystyle \frac{\sin^3(x)}{\cos(x)}​

Then change sin³(x) to (sin(x))(1-cos²(x))

The integrand becomes:

\displaystyle \frac{\sin(x)}{\cos(x)}-\sin(x)\cos(x)​