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)
∫(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...