# Wien's fifth power Law and Stephan Boltzmann's fourth power laws of emissive power

Wien's fifth power law says that emissive power is proportional to the temperature raised to the fifth power. On the other hand, the Stefan–Boltzmann law says emissive power is proportional to the temperature raised to the fourth power. How can both of these be true?

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From Wien's displacement law (derived here), $\lambda_\max T = b$ is constant. So, from Planck's law (e.g. see here):
$I'(\lambda_\max, T) \propto \frac{\lambda_\max^{-5}}{e^{hc/(\lambda_\max k T)} -1} = \frac{T^5/b^5}{e^{hc/(k b)} -1} \propto T^5$
and integrating Plank's law (e.g. here except I keep in terms of wavelength for slightly longer) we get, by changing variables to $x=hc/(\lambda kT)$ so that $\lambda^{-5}d\lambda \propto -T^4x^3dx$ (the negative is absorbed by changing the integration interval limits):
$\int_0^\infty I'(\lambda,T)d\lambda \propto \int_0^\infty \frac{T^4 x^3dx}{e^{x}-1} \propto T^4$.