# Substitution method with Integration by Parts?

1. Jan 17, 2012

### csinger1

Substitution method with Integration by Parts???

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

Evaluate the integral...
∫x^3[e^(-x^2)]dx

2. Relevant equations

∫udv=uv-∫vdu

3. The attempt at a solution
I first tried using integration by parts setting u and dv equal to anything and everything. This seemed to make it more complicated so I decided to use the substitution method setting y=-x^2 and dy=-2xdx and finally -1/2dy=xdx to give me
1/2∫ye^ydy
From here i used integration by parts with u=y, du=dy, v=e^y and dv=e^y dy to get
1/2(ye^y-∫e^y dy) = 1/2(ye^y-e^y)
and then
1/2[-x^2e^(-x^2)-e^(-x^2)]
This problem got really confusing with all the variables and I just want to make sure that I'm not way off base with my methods and solution.
Thanks in advance for any input!

2. Jan 17, 2012

### Dick

Re: Substitution method with Integration by Parts???

Looks right to me.

3. Jan 17, 2012

### csinger1

Re: Substitution method with Integration by Parts???

That's a relief! I spent a loooonnnggg time working this problem. Thanks!

4. Jan 17, 2012

### Staff: Mentor

Re: Substitution method with Integration by Parts???

Once you have gotten an antiderivative, it's a good practice to check it by differentiating it. If your work is correct, the derivative of your answer should be the original integrand.