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Math Help - Integration by Parts Question

  1. #1
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    Integration by Parts Question

    Let be a positive integer. We define .
    Using integration by parts determine the constant such that .



    What is the value of p?
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  2. #2
    MHF Contributor red_dog's Avatar
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    I_{n+3}=\int_0^{\infty}e^{-3u^3}u^{n+3}du=-\frac{1}{9}\int_0^{\infty}\left(e^{-3u^3}\right)'u^{n+1}du=

    =\left.-\frac{1}{9}\cdot\frac{u^{n+1}}{e^{3u^3}}\right|_0^  {\infty}+\frac{n+1}{9}I_n=\frac{n+1}{9}I_n

    Then (n+1)I_n=9I_{n+3}\Rightarrow p=9
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