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Math Help - Strange question with improper integrals

  1. #1
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    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?
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  2. #2
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    Write it as  \lim_{b \to \infty} \int^{b}_{-b} x^{5}e^{-x^{6}} \ dx and make the substitution  u = -x^{6}
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    Quote Originally Posted by Archduke01 View Post








    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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