Gaussian Integral Simplification

[superspartan9]

Homework Statement

The integral of (x^n)(e^(-a*x^2)) is easier to evaluate when n is odd.
a) Evaluate ∫(x*e^(-a*x^2)*dx) (No computation allowed!)
b) Evaluate the indefinite integral of x*e^(-a*x^2), using a simple substitution.
c) Evaluate ∫(x*e^(-a*x^2)*dx) [from o to +inf]
d) Differentiate the previous result to evaluate ∫((x^3)(e^(-a*x^2))dx)

Homework Equations

∫(e^(-a*x^2)*dx) = (1/2)√(∏/a)

The Attempt at a Solution

I thought that the easiest solution might be using integration by parts, but I ran into the issue of the range being different on the integrals, and I have no idea how else I can do this unless I can assume that the function is even...

 

[vela]

What are the limits on the integrals?

 

[superspartan9]

The first one is -inf to inf, second is indefinite, third is 0 to inf, and fourth is 0 to inf.

 

[Dick]

The first one is -inf to inf, second is indefinite, third is 0 to inf, and fourth is 0 to inf.

You don't need integration by parts for any of those. Take them one at a time. Start with a). The integrand isn't even, it's odd.

 

[superspartan9]

Alright, I figured out the first one in class by drawing the graph for x and e^(-ax^2) and realized that the x made the function odd, as you said, and that the integral was then 0 for -inf to inf.

I'm going to try and tackle c) and d) because I have no idea where to start on b) with that substitution. My brain is stuck on integration by parts >_> if you could give me a hint or something as to where to begin with the indefinite integral, that would be great!

 

[superspartan9]

Update: Just tried substituting u = a*x^2 into the integral, and it evaluated to -(1/2a)*e^(-a*x^2)

I think it's right... but I'm not sure, the substitution worked though because the du = 2ax dx which means we can just throw the constants in there.

 

[superspartan9]

Another Update: The above equation is correct because it yields the correct value for part c).

Now I'm just stuck on d) where it's asking me to differentiate... More later if I figure it out.

 

[vela]

Try differentiating it and see if you recover the integrand.

 

[superspartan9]

Got it! Taking the derivative with respect to a of part c) yields the answer! :D