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I was told that $$ \int_0^{\infty}\frac{x^3}{e^x-1}dx=\frac{\pi^4}{15}$$ in physics class.

I have no idea to compute this integral by hand. Could someone help me?

lmksdfa
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  • Did you get the answer you wanted? If not I actually made a page all about this and can explain more if needed. Contact me if you want better insight into this integral. Thanks! – Highvoltagemath Aug 12 '19 at 22:13

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Expand $$\frac{x^3}{e^x-1}=\frac{x^3e^{-x}}{1-e^{-x}} =\sum_{n=1}^\infty x^3e^{-nx}$$ and then integrate termwise.

Angina Seng
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