Planck Black-Body Law

1. The problem statement, all variables and given/known data

Starting from the Planck-Body Law



I[itex]_{λ}[/itex]dλ = [itex]\frac{2\pi c^{2}h}{λ^{5}}[/itex] [itex]\frac{1}{e^{hc/(λkT)} - 1}[/itex]dλ



where λ is the wavelength, c is the speed of light in a vaccuum, T is the temperature, k is Boltzmann’s constant,

and h is Planck’s constant, prove that the total energy density over all wavelengths is given by



I[itex]_{tot}[/itex] = aT[itex]^{4}[/itex]



and express a in terms of pi,k,h,c

2. Relevant equations

λ = c/f





3. The attempt at a solution

Our teacher gives us a hint "think about whether it is better to do the integral in the wavelength or frequency domain" - which in this case means he wants us to switch to the frequency domain. I did try a bunch of things but I am just not sure if my first step is correct. To switch to the frequency domain, all I havr to do is plug in



λ = c/f

and

dλ = -c/f[itex]^{2}[/itex]



correct? Or is this first step wrong