$\zeta(4)=\sum_ {k=1}^{\infty}{\frac{1}{k^4}}$ [duplicate]

How to Find $$\zeta(4)=\sum_ {k=1}^{\infty}{\frac{1}{k^4}}$$ the most basic way possible?
I know it's $\pi^4/90$ but to arrive at this figure? Curious, because I need it to solve the integral $\int_{0}^{\infty}x^3\cdot(e^x-1)^{-1}\;dx$.