# Strange question with improper integrals

• Apr 29th 2010, 06:49 PM
Archduke01
Strange question with improper integrals
http://euler.vaniercollege.qc.ca/web...fc203244b1.png

First off, the limits of integration are both infinites, and I've only worked with one so far, with the other being a number. What changes?

And finally, the exponent within the exponent of e throws me off. There's a way to integrate that entire thing using substitution, but I'm not sure how. Can anyone tell me or at least link me to some tutorial online that explains it?
• Apr 29th 2010, 06:54 PM
Random Variable
Write it as $\displaystyle \lim_{b \to \infty} \int^{b}_{-b} x^{5}e^{-x^{6}} \ dx$ and make the substitution $\displaystyle u = -x^{6}$
• Apr 29th 2010, 07:01 PM
tonio
Quote:

Originally Posted by Archduke01

First off, the limits of integration are both infinites, and I've only worked with one so far, with the other being a number. What changes?

And finally, the exponent within the exponent of e throws me off. There's a way to integrate that entire thing using substitution, but I'm not sure how. Can anyone tell me or at least link me to some tutorial online that explains it?

As $\displaystyle (-x^6)'=-6x^5$ , this is a very easy integral:

$\displaystyle \int\limits^\infty_{-\infty}x^5\,e^{-x^6}\,dx=$ $\displaystyle \lim_{a\to\infty\,,\,b\to -\infty}-\frac{1}{6}\int\limits^a_b-6x^5\,e^{-x^6}\,dx$ $\displaystyle =\lim_{a\to\infty\,,\,b\to -\infty}-\frac{1}{6}\,\left[e^{-x^6}\right]^a_b = 0$ , which shouldn't be too surprising since the function is odd...(Wink)

Tonio