An Introduction to Thermal Physics · Problem B.21

Systems of Interacting Particles — Daniel Schroeder

Physics — Statistical Mechanics · fermi-gas-integral

Problem

In calculating the heat capacity of a degenerate Fermi gas in Section 7.3, we needed the integral

\int_{-\infty}^{\infty} \frac{x^2 e^x}{(e^x+1)^2}\, dx = \frac{\pi^2}{3}.

To derive this result, first show that the integrand is an even function, so it suffices to integrate from 0 to

\infty

and then multiply by 2. Then integrate by parts to relate this integral to the one in equation B.35.