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Math Help - Integral help needed

  1. #1
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    Integral help needed

    I_{n} = \int_{0}^{\infty}\frac{ln(x)}{n^2 + x^2}dx

    I tried going by parts and using arctan but I get stuck after there, can anyone point me in the right direction please?

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  2. #2
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    Quote Originally Posted by kevinlightman View Post
    I_{n} = \int_{0}^{\infty}\frac{\ln x}{n^2 + x^2}dx

    I tried going by parts and using arctan but I get stuck after there, can anyone point me in the right direction please?

    (Step 1)  \text{ Let } t=\frac{n}{x} \Rightarrow I_n = \frac{1}{n} \int_0^\infty \frac{\ln n - \ln t} {1 + t^2} \; dt=\frac{\pi \ln n}{2n} - \frac{1}{n}\int_0^\infty \frac{ \ln t} {1 + t^2} \; dt

    (Step 2) \int_0^\infty \frac{ \ln t} {1 + t^2} \; dt = \int_0^1 \frac{ \ln t} {1 + t^2} \; dt + \int_1^\infty \frac{ \ln t} {1 + t^2} \; dt  = \int_0^1 \frac{ \ln t} {1 + t^2} \; dt - \int_0^1 \frac{ \ln t} {1 + t^2} \; dt = 0

    (Step 3)  I_n = \frac{\pi \ln n}{2n}
    Last edited by mr fantastic; September 18th 2009 at 09:29 AM. Reason: Restored original reply
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  3. #3
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    Substitution x=nt will solve the problem.

    (Ahhh, luobo beat me to it.)
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