Tricky Integration with Trig Substitution

[bjohnson2001]

Homework Statement

Evaluate.

\int(4-y)\sqrt{4-y^{2}}dy

I have the solution using CAS software here:

2y\sqrt{4-y^{2}}+8sin^{-1}\frac{y}{2}+\frac{1}{3}(4-y^{2})^{3/2}


but I need to do this by hand. I have researched the usual trig methods but am having some difficulty. Can someone please help me find the right identity?

 

[Dick]

Substituting y=2*sin(t) looks like a good starting point.

 

[bjohnson2001]

I don't think it goes anywhere though, I get:

8cos(t)-4sin(t)cos(t)

when it comes time to bring the original variable back in I get a mess of cos(arcsin(y/2)) but maybe I am missing something

 

[Dick]

I don't think it goes anywhere though, I get:

8cos(t)-4sin(t)cos(t)

when it comes time to bring the original variable back in I get a mess of cos(arcsin(y/2)) but maybe I am missing something

If y=2*sin(t) then dy=2*cos(t)dt. I think you are forgetting that. And yes, it does take some work to integrate. If you make a try at it and show your steps I'm sure someone will try to help.