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Math Help - Without using Integral by Part

  1. #1
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    Without using Integral by Part

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  2. #2
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    Re: Without using integral by parts

    \displaystyle\int_0^{\infty}e^{-t}\arctan t\;dt=\int_0^{\infty}\int_0^t \frac{e^{-t}}{1+x^2} \;dx\;dt

    Now switch the order of integration:

    \qquad\qquad=\int_0^{\infty}\int_x^{\infty} \frac{e^{-t}}{1+x^2} \;dt\;dx=\int_0^{\infty}\frac{e^{-x}}{1+x^2}\;dx
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  3. #3
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    Re: Without using integral by parts

    Quote Originally Posted by LordoftheFlies View Post
    \displaystyle\int_0^{\infty}e^{-t}\arctan t\;dt=\int_0^{\infty}\int_0^t \frac{e^{-t}}{1+x^2} \;dx\;dt

    Now switch the order of integration:

    \qquad\qquad=\int_0^{\infty}\int_x^{\infty} \frac{e^{-t}}{1+x^2} \;dt\;dx=\int_0^{\infty}\frac{e^{-x}}{1+x^2}\;dx

    Can u explain that step?

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  4. #4
    Newbie LordoftheFlies's Avatar
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    Re: Without using Integral by Part

    It's a standard trick. Watch this video, he'll explain the basic idea better than I can.

    Spoiler:
    Last edited by LordoftheFlies; January 2nd 2013 at 04:32 AM.
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