Strange question with improper integrals

**Archduke01** · April 29th 2010, 06:49 PM

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?

**Random Variable** · April 29th 2010, 06:54 PM

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

**tonio** · April 29th 2010, 07:01 PM

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

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

Tonio

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