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Suggested textbook for techniques of integration?

  1. Feb 14, 2006 #1
    In my recent reading, I find myself stumbing across integral identities such as
    \int_{0}^{\infty} \frac{x dx}{e^{ax} + 1} = \frac{\pi^2}{12a^2}
    \int_{0}^{2\pi} \frac{cos(\theta) d\theta}{A + B cos(\theta)} = \frac{2\pi}{B}(1 - \frac{A}{\sqrt{A^2 - B^2}})

    Can anyone recommend a textbook that would assist me in tackling identities such as these?

    (Parenthetically, am I wrong in thinking that these are slightly beyond the ordinary Calculus I techniques of integration? I suspect that the first can be tackled by contour integration, and progress on the second might be possible using a substitution such as [tex]u = tan(\theta / 2)[/tex], but neither one is exactly clear to me, which is why I think a good textbook would be helpful.)

    Thanks in advance.
  2. jcsd
  3. Feb 14, 2006 #2


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    Although it may sound strange, these kind of 'real integrals' can often be determined easily using complex analysis (residue calculation) which indeed involves contour integration.
  4. Feb 14, 2006 #3
    Yeah, I know. It seems especially appropriate here, since a search for [tex]\pi^2/12[/tex] turns up that it equals [tex]\sum_{n=0}^\infty \frac{(-1)^n}{n^2}[/tex] and there is such a lovely sequence of poles sitting on the imaginary axis. I suspect that the contour is the positive real axis, a quarter circle over to the imaginary axis, and then that axis with indentations to avoid the poles. I vaguely recall some indentation lemma that gives the contribution of each pole in terms of the residue and how far around you are going. I'm unclear on many of the details though...

    Which brings me back to my original point... is there some sort of advanced integration techniques textbook that would clarify these things and others? (A complex analysis text would help with this problem, but I'm not convinced it would help with the other integral I mentioned... so I'd prefer a book that focused on solving more difficult integrals.)
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