# Tricky Integration with Trig Substitution

1. Feb 21, 2012

### bjohnson2001

1. The problem statement, all variables and given/known data

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?

2. Feb 21, 2012

### Dick

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

3. Feb 21, 2012

### 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

4. Feb 21, 2012

### Dick

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.