# Integral Representation of Infinite series

Let's take a look at the following integrals :

1) $\displaystyle \int\limits_{0}^{1} \frac{\log{x}}{1+x} \ dx = -\frac{\pi^{2}}{12} = -\frac 1 2 \sum\limits_{n=1}^{\infty} \frac{1}{n^2}= -\frac 1 2 \zeta(2)$

2) For $c<1$ $\displaystyle \int\limits_{0}^{\frac{\pi}{2}} \arcsin(c \cos{x}) \ dx = \frac{c}{1^2} + \frac{c}{3^2} + \frac{c}{5^2} + \cdots$

I have seen integral representations of series and sums employed in ingenious way ways to to compute closed-forms and deduce other interesting properties (e.g. asympotics, recurrences, combinatorial interpretations, etc). Are there any general algorithms or theories behind such methods of integral representations?

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The question is now re-opened and had been edited significantly from its original form. Hence off-topic or obsoleted comments have been deleted. –  Willie Wong May 2 '12 at 8:00
For further detail see these meta threads on the question closing and reopening. –  Bill Dubuque May 29 '12 at 17:56