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Math Help - Integration proof

  1. #1
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    Integration proof

    Which substitution can I use to help me prove that the integral of ln (x) / (1 + x^2) between 0 and 1 is identical to -1 * the same integral between 1 and infinity?
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  2. #2
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    Quote Originally Posted by kevinlightman View Post
    Which substitution can I use to help me prove that the integral of ln (x) / (1 + x^2) between 0 and 1 is identical to -1 * the same integral between 1 and infinity?
    The first thing I notice is that the substitution u= 1/x changes x= 0 to 1 into u= infinity to 1. du= -\frac{1}{x^2}dx so dx= -x^2 du= -\frac{1}{u^2}du. The only "tricky" part is recognizing the x^2+ 1= \frac{1}{u^2}+ 1= \frac{u^2+ 1}{u^2} and that is not all that "tricky"!
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    The first thing I notice is that the substitution u= 1/x changes x= 0 to 1 into u= infinity to 1. du= -\frac{1}{x^2}dx so dx= -x^2 du= -\frac{1}{u^2}du. The only "tricky" part is recognizing the x^2+ 1= \frac{1}{u^2}+ 1= \frac{u^2+ 1}{u^2} and that is not all that "tricky"!
    Great thanks, you're right it does seem a lot less tricky now I can see it working
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  4. #4
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    Sorry edited double post
    Last edited by kevinlightman; August 30th 2009 at 03:08 AM.
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