# Thread: Strange question with improper integrals

1. ## Strange question with improper integrals

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?

2. Write it as $\displaystyle \lim_{b \to \infty} \int^{b}_{-b} x^{5}e^{-x^{6}} \ dx$ and make the substitution $\displaystyle u = -x^{6}$

3. 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...

Tonio