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Simplify result of this definite integral

asked 11 years ago

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It is well known that for nN and n>0 (an maybe even for more than these restrictions): In=0xnex1dx=ζ(n+1)n!

which, analytically can be shown easily by expanding 1/(1ex) into a geometric series, which leads to trivial integrals, and by using ζ(n+1)=l=1l(n+1). So, eg.: I1=π2/6
I2=2ζ(3)
a.s.o...

Now, if I try even the simplest case with sage, I get this 'nifty' little results

sage: integrate(x/(exp(x)-1),x,0,oo)

-1/6*pi^2 + limit(-1/2*x^2 + x*log(-e^x + 1) + polylog(2, e^x), x,
+Infinity, minus)

Is there any trick to simplify this down to the final result, or is this about as far as I can get with sage alone?

PS.: it is probably needless to say that (once again ... :( ...)

In[1]:= Integrate[x/(Exp[x] - 1), {x, 0, Infinity}]
Out[1]:= Pi^2/6
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answered 9 years ago

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for an answer to this and related other questions see http://thingwy.blogspot.de/

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That's not an anwer fitting in this forum.

rws gravatar imagerws ( 9 years ago )

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Asked: 11 years ago

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Last updated: Aug 08 '15