a
Let's consider a strip of thickness \(dx\) at a distance of \(x\) from the left end as shown in the figure.

\(\frac{y}{x}=\frac{d}{a}\)

\(\Rightarrow \quad y=\left(\frac{d}{a}\right) x\)

\(C_{1}=\frac{\varepsilon_{0} a d x}{(d-y)} \quad ; \quad C_{2}=\frac{k \varepsilon_{0} a d x}{y}\)

\(C_{e q}=\frac{C_{1} C_{2}}{C_{1}+C_{2}}=\frac{k \varepsilon_{0} a d x}{k d+(1-k) y}\)

Now integrating it from \(0\) to \(a\)

\(\int_0^a {\frac{{{\text{k}}{\varepsilon _0}{\text{adx}}}}{{{\text{kd}} + (1 - {\text{k}})\frac{{\text{d}}}{{\text{a}}}{\text{x}}}}}  = \frac{{{\text{k}}{\varepsilon _0}{{\text{a}}^2}{\text{lnk}}}}{{{\text{d}}({\text{k}} - 1)}}\)