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Math Help - Gamma fuction and integral

  1. #1
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    Exclamation Gamma fuction and integral

    Hello.

    I just cant solve this problem. Who can do it?

    Using gamma fuction calculate the following integral:
    <br />
\int_o^1 x^3 [ln(\frac{1}{x})]^3 dx<br />

    Thanks a lotĦĦ
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by osodud View Post
    Hello.

    I just cant solve this problem. Who can do it?

    Using gamma fuction calculate the following integral:
    <br />
\int_o^1 x^3 [ln(\frac{1}{x})]^3 dx<br />

    Thanks a lotĦĦ
    Let u=\ln\!\left(\tfrac{1}{x}\right)\implies \,du=-\tfrac{1}{x}\,dx. Note that u=\ln\!\left(\tfrac{1}{x}\right)\implies e^{-u}=x

    Also note that u(0)\rightarrow\infty and u(1)=0.

    Therefore, \int_0^1x^3\left[\ln\!\left(\tfrac{1}{x}\right)\right]^3\,dx\xrightarrow{u=\ln\!\left(\tfrac{1}{x}\right  )}{} \int_{\infty}^{0}-x^4 u^3\,du=\int_0^{\infty}e^{-4u}u^3\,du

    Now make the substitution t=4u\implies \,dt=4\,du. Also note that t(0)=0 and t(\infty)=\infty

    Therefore \int_0^{\infty}e^{-4u}u^3\,du\xrightarrow{t=4u}{}\tfrac{1}{4}\int_0^{  \infty}e^{-t}\left(\tfrac{t}{4}\right)^{4}\,dt=\tfrac{1}{1024  }\int_0^{\infty}e^{-t}t^{4}\,dt

    Now apply the definition of the gamma function to get \tfrac{1}{1024}\int_0^{\infty}e^{-t}t^{4}\,dt=\frac{\Gamma\!\left(5\right)}{1024}=\f  rac{24}{1024}=\frac{3}{128}

    Does this make sense?
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