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\(\displaystyle \int_{0}^\infty \frac{x^3}{e^x-1} \text{d}x\)

It looked like this integral would best be handled by treating it as complex-valued and setting up a contour so that the complex parts dropped out. However, the students did not know any complex analysis and were looking for a more elementary approach. It wasn't obvious how to do it using real integrals (and I tried to use tricks as in the evaluation of \(\displaystyle \int_{-\infty}^\infty e^x \text{d}x\)). I also discussed this with another graduate student, and we were actually even unable to select a decent contour to take the integral. (Though we did put it into Mathematica which computed \(\displaystyle \frac{\pi^4}{15}\).) We're basically baffled. I was hoping to find some advice about this integral here. Any help would be appreciated!