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Math Help - Improper integrals:

  1. #1
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    Improper integrals:

    Im stuck with this problem:
    \int_{-\infty}^{+\infty} xe^{-x^2}dx
    kindly help
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  2. #2
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    Use that \int_{ - \infty }^\infty {f(x)\,dx} = \mathop {\lim }\limits_{a \to - \infty } \int_a^c {f(x)\,dx} + \mathop {\lim }\limits_{b \to \infty } \int_c^b {f(x)\,dx}
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    Im stuck with this problem:
    \int_{-\infty}^{+\infty} xe^{-x^2}dx
    kindly help
    Another hint. Let y = x^2.

    -Dan
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  4. #4
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    -\frac{e^{-x^2}}{2}\big|_{-\infty}^{0} +(- \frac{e^{-x^2}}{2})\big|_{0}^{\infty}

    the answer is zero?
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  5. #5
    MHF Contributor red_dog's Avatar
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    \displaystyle\int_{-\infty}^{\infty}xe^{-x^2}dx=\lim_{a\to\infty}\int_{-a}^axe^{-x^2}dx
    f(x)=xe^{-x^2} is an odd function and [-a,a] is a symmetric interval, so the integral is 0, thus the limit is 0.
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