A parallel plate capacitor has plates of area A separated by distance ‘d’ between them. It is filled with a dielectric which has a dielectric constant that  varies as  kx=K01+αx where ‘x’ is the distance measured from one of the plates. If  αd<<1, the total capacitance of the system is best given by the expression

Capacitance of “dx” element  C'=Kε0Adx Capacitance of “dx” element  C'=K01+αxε0AdxAll such elements are connected in series. Hence, net capacitance is given by, ∑1C'=∫0ddxK0ε0A1+αx               ⇒1C=1K0ε0Aαln1+αx0b∵∫1axdx=logaxa+c ⇒1C=1K0ε0Aαln1+αd     Given    αd<<1    ⇒1C=1K0ε0Aααd−α2d22      [using ln(1+x) = x-x22+...]⇒1C=dK0ε0A1−αd2                            ⇒C=K0ε0Ad1-αd2-1 ⇒C=K0ε0Ad1+αd2         [using    (1+x)n  = 1+nx  ]