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I mean the Laurent series at $s=1$.

I want to do it by proving $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$,

based on the integral formula given in Wikipedia. But I cannot solve this integral except by using Mathematica. Tried complex analytic ways but no luck. Any suggestions? Thanks for your attention!

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    You can find a possible approach at [this posting](http://sos440.tistory.com/251) in my blog.2012-03-23
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    Related: http://math.stackexchange.com/questions/100790/limit-of-zeta-function2012-03-23
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    Thanks a lot, @sos440 and David! So I know several ways solving the problem in the title now.2012-03-23

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