A ring of radius R carries a non-uniform charge of linear density $\mathrm{\lambda }={\mathrm{\lambda }}_0\mathrm{cos\theta }$ (see in the figure). Magnitude of the net dipole moment of the ring is

Charge, dq$={\lambda }_{0}R\cos \theta \cdot d\theta$ 

dipole moment$={\lambda }_0\cos \theta \mathrm{Rd}\theta \times 2\mathrm{Rcos}\theta$
$=2{\lambda }_0{\mathrm{R}}^2{\cos }^2\theta \mathrm{d}\theta$ 

The net electric dipole moment is given as,

$$\begin{gathered} \overrightarrow{\mathrm{p}}=\int \overrightarrow{\mathrm{r}}\mathrm{dq} \\ ={\int }_0^{2\mathrm{\pi }} {\mathrm{\lambda }}_0\cos \mathrm{\theta Rd\theta } \end{gathered}$$

$\begin{array}{r}\overrightarrow{\mathrm{p}}={\int }_0^{2\mathrm{\pi }} {\mathrm{\lambda }}_0{\mathrm{R}}^2{\cos }^2\mathrm{\theta d\theta }\\ =\frac{{\mathrm{\lambda }}_0{\mathrm{R}}^2}{2}{\int }_0^{2\mathrm{\pi }} (1+\cos 2\mathrm{\theta })\mathrm{d\theta }\\ =\frac{{\mathrm{\lambda }}_0{\mathrm{R}}^2}{2}{\left[\mathrm{\theta }+\frac{1}{2}\sin 2\mathrm{\theta }\right]}_0^{2\mathrm{\pi }}\\ ={\mathrm{\lambda }}_0{\mathrm{R}}^2\mathrm{\pi }\end{array}$