$\frac{y}{x}=\frac{d}{a}$

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

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

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

Now integrating it from $0$ to $a$

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