For a doublet,the  stream function is $$\psi=\frac{-k}{2\pi}\frac{\sin\theta}{r}$$

$$V_{r}=\frac{\partial \phi}{\partial r}=\frac{1}{r}\frac{\partial \psi}{\partial \theta}$$

Here $$\psi=\frac{-k}{2\pi}\frac{\sin\theta}{r} \\\Rightarrow \frac{\partial \psi}{\partial \theta}=\frac{-k}{2\pi}\frac{\cos\theta}{r} \\\frac{\partial \phi}{\partial r}=\frac{1}{r}\left ( \frac{-k}{2\pi}\frac{\cos\theta}{r} \right )=\frac{-k}{2\pi}\frac{cos\theta}{r^{2}}$$

On integrating with respect to ‘r’

$$\phi=\left ( \frac{-k}{2\pi}cos\theta \right )\left ( -\frac{1}{r} \right ) \\\Rightarrow \phi=\frac{k}{2\pi}\frac{\cos\theta}{r}$$