Solving Stat Mech Integral with Wolfram Alpha

In summary, the conversation discusses the evaluation of an integral of the form ##\int_0^{\infty} \frac{p^2 dp}{\mathrm{exp}(a\sqrt{p^2+b^2}) \pm 1}## and the use of a change of variables ##x(p) = a\sqrt{p^2 + b^2}## to simplify the integral. The resulting integral is ##\frac{1}{a^3} \int_0^{\infty} \frac{\sqrt{x^2-(b/a)^2}}{e^x \pm 1} dx##. Wolfram Alpha is not helpful in finding a solution, and the conversation mentions the
  • #1
ergospherical
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Looking to evaluate an integral of the form $$\int_0^{\infty} \frac{p^2 dp}{\mathrm{exp}(a\sqrt{p^2+b^2}) \pm 1} $$Changing to ##x(p) = a\sqrt{p^2 + b^2}## gives $$\frac{1}{a^3} \int_0^{\infty} \frac{\sqrt{x^2-(b/a)^2}}{e^x \pm 1} dx$$Wolfram alpha doesn't tell me anything useful, sadly.
 
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  • #3
ergospherical said:
Looking to evaluate an integral of the form $$\int_0^{\infty} \frac{p^2 dp}{\mathrm{exp}(a\sqrt{p^2+b^2}) \pm 1} $$Changing to ##x(p) = a\sqrt{p^2 + b^2}## gives $$\frac{1}{a^3} \int_0^{\infty} \frac{\sqrt{x^2-(b/a)^2}}{e^x \pm 1} dx$$Wolfram alpha doesn't tell me anything useful, sadly.
You might want to double check your algebra for changing variables (e.g. do the bounds of the integral over ##x## make sense?)
 

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