Question

In the 18.01 practice questions for Exam 4 problem 3a, we are asked to express int[x^2 * e^-x^2] in terms of int[e^-x^2].

I understand that this should involve using integration by parts but the given solution doesn't show working and I can't obtain the same answer. Can someone help?

Link to the problem set:


http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/exams/prexam4a.pdf

Answer 1

This was answered on the math.stackexchange site here:

http://math.stackexchange.com/questions/49520/express-int1-0x2-e-x2-dx-in-terms-of-int1-0e-x2-dx

Answer 2

Integration by parts works if you let
\[u = x^{2n - 1}, dv = e^{-x^{2}} dx.\]
You end up with
\[h + k \int\limits_{0}^{1}(x^{2n - 2}e^{-x^{2}} dx,\]
where h and k are the constants you get when you carry out all the calculations involved in doing the integration by parts. The complete answer does not show up in the attachment.

Answer 3

I send you a note with the solution