Wien's law
$$\frac{d}{d\lambda} I_{\lambda}(\lambda,T)=0$$
$$\frac{-5}{\lambda^6}\frac{1}{\exp\left(\frac{hc}{\lambda kT}\right)-1}+\frac{ hc }{kT\lambda^7}\frac{\exp\left(\frac{hc}{\lambda kT}\right)}{\left(\exp\left(\frac{hc}{\lambda kT}\right)-1\right)^2}=0$$
$$e^x(x-5)+5=0$$
$$x=\frac{hc}{\lambda_{maxI}kT}\approx 4.965$$
$$\lambda_{maxI}T\approx 2.9\cdot 10^{-3}K\cdot m$$

Stefan-Boltzmann's law
$$\int_{0}^{\infty}I(\nu,T)d\nu=\int_{0}^{\infty}\frac{2\pi\nu^2}{c^2}\frac{h\nu}{\exp\left(\frac{h\nu}{kT}\right)-1}d \nu=$$
$$=\frac{2\pi k^4 T^4}{h^3c^2}\int_{0}^{\infty}\frac{x^3}{\exp\left(x\right)-1}dx$$

$$\int_{0}^{\infty}\frac{x^3}{\exp\left(x\right)-1}d x=\frac{\pi^4}{15}$$

$$\sigma=\frac{2\pi^5 k^4}{15 h^3 c^2}\approx 5.67\cdot 10^{-8} \frac{W}{m^2K^4}$$
$$\int_{0}^{\infty}I(\nu,T) d\nu=\sigma T^4$$