Different versions of Planck's law

Question

For a presentation in physics I am going to talk about black body radiation since our book just mentions Planck's law, Wien's displacement law and Stefan-Boltzmann's law. I want to derive Wien's law and Stefan-Boltzmann's law from Planck's law. I have managed to derive Stefan-Boltzmann's law from: $$u(\nu)=\frac{2\pi h \nu^3}{c^2}\frac{1}{e^\frac{h\nu}{kT}-1}$$ by integrating over all $\nu$ from $0$ to $\infty$. However, my textbook says: $$u(\lambda,T)=\frac{2\pi hc^2 }{\lambda^5}\frac{1}{e^\frac{hc}{\lambda kT}-1}$$ If I can get from the second formula to the first, I will be happy alltough, a simple and informal derivation of Plack's law would be very satisfying. I also managed to derive Wien's law by derivation and setting equal to zero by using: $$u_\lambda=\frac{8\pi hc }{\lambda^5}\frac{1}{e^\frac{hc}{\lambda kT}-1}$$ What are the differences? And please, since I am Norwegian and do not know very advanced physics, do not use any advanced expression without explaining them. Thanks!

Answer 1

First is given in terms frequency (nu) and second is given in terms of wavelength(lambda). Both are same. If you make substituion nu=c/lambda , you can easily get the second relation.