The gap between the plates of a parallel plate capacitor of area A...

$\dfrac{1}{dC} = \dfrac{dx}{{ϵ}_{0}ϵA}$

$= \dfrac{dx}{{ϵ}_{0}\left[{ϵ}_{1} + \dfrac{\left({ϵ}_{2} - {ϵ}_{1}\right)}{d}x\right]A}$

$\dfrac{1}{C} = ∫ \dfrac{1}{dC} = \dfrac{1}{{\in}_{0}A}\mathop{∫}_{0}^{x}\dfrac{dx}{{\in}_{1} + \left(\dfrac{{\in}_{2} - {\in}_{1}}{d}\right)}$

$= \dfrac{\ln \left({\in}_{2} / {\in}_{1}\right)d}{{\in}_{0}A\left({\in}_{2} - {\in}_{1}\right)}$

$C = \dfrac{{\in}_{0}A\left({\in}_{2} - {\in}_{1}\right)}{\ln \left({\in}_{2} / {\in}_{1}\right) \cdot d}$