can the prime really not solve this integral?

[parisse]

(02-27-2014 09:11 PM)Manolo Sobrino Wrote:  

(02-27-2014 12:21 PM)parisse Wrote:  You get a numeric answer with:
int(x^3/(exp(x)-1),x,0,inf)
then shift-enter.
x^3/(exp(x)-1) does not have an antiderivative than you can express with elementary function (special functions required, polylogs here). There is probably a trick than can give you the exact answer (pi^4/15) for the definite integral, any idea? On a voyage 200, you don't get the exact value by the way.

No, no, you don't use primitives for this, come on.
You could use contour integration/Residue Theorem, but it's a bit tricky for this. See for instance:

http://en.wikipedia.org/wiki/Stefan_bolt...w#Appendix

The easy way is:

http://math.stackexchange.com/questions/...u3-eu-1-du

That's exactly what I meant by a trick to find the exact answer for the definite integral.

Quote:(Now that I'm thinking about it, it shouldn't be too difficult to implement a large class of definite integrals by the Residue theorem. That would be come in handy.)

There is already a class of definite integrals that are handled by the residue theorem. But this one requires a trick before one can apply it, I can not implement that. Expanding 1/(exp(x)-1) is a better trick, but I don't know if that can lead to something sufficiently general (I don't like tables). And it's not that important anyway, since the numeric answer is reliable.