Zeta function for Debye Low Temp. Limit

[phys_student1]

Hello,

In the low temp. limit of Debye law for specific heat, we encounter the following integral:

∫(x⁴ e^x)/(e^x-1)² dx, from 0 to ∞.

The result is 4∏²/15.

I have searched and found this to be related to zeta function but zeta functions do not have e^x in the numerator so I am unable to evaluate the integral using them.

Any help?

 

[phys_student1]

Never mind. I found that what should be integrated is the energy, ∫x³/(e^x-1) which can be integrated easily using integration by parts then using zeta function, the result is π²/15.