Tricky Derivation of Blackbody Equations.

[omegas]

Homework Statement

Starting with the Planck distribution:

R(\lambda,T) = \frac{c}{4} \frac{8 \pi}{\lambda^4} (\frac{hc}{\lambda})(\frac{1}{e^{hc/(\lambda kT)}-1})

Derive the blackbody Stefan-Boltzmann law (ie total flux is proportional to T⁴) by integrating the above expression over all wavelengths. Thus show that

R(T) = \frac{2 \pi^5 k^4}{15h^3 c^2} T^4

and \int \frac{x^3}{e^x -1} dx = \frac{\pi^4}{15}

The Attempt at a Solution

I know I need to substitute x = \frac{hc}{kT} \frac{1}{\lambda}. And somehow I think I can use the form KR(\lambda,T) = A(\lambda)B(\lambda)

 

[zachzach]

Homework Statement

Starting with the Planck distribution:

R(\lambda,T) = \frac{c}{4} \frac{8 \pi}{\lambda^4} (\frac{hc}{\lambda})(\frac{1}{e^{hc/(\lambda kT)}-1})

Derive the blackbody Stefan-Boltzmann law (ie total flux is proportional to T⁴) by integrating the above expression over all wavelengths. Thus show that

R(T) = \frac{2 \pi^5 k^4}{15h^3 c^2} T^4

and \int \frac{x^3}{e^x -1} dx = \frac{\pi^4}{15}

The Attempt at a Solution

I know I need to substitute x = \frac{hc}{kT} \frac{1}{\lambda}. And somehow I think I can use the form KR(\lambda,T) = A(\lambda)B(\lambda)

Just use that substitution and try to get it into the form of the integral provided.