Suppose $u$ is a twice continuously differentiable real-valued harmonic function on a disk $D(z_0;r)$ centered at $z_0 = x_0 +iy_0$. For $(x_1, y_1) \in D(z_0;r)$, show that the equation
\begin{align*} v(x_1, y_1) = c + \int_{y_0}^{y_1} \frac{\partial{u}}{\partial{x}}(x_1,y)dy-\int_{x_0}^{x_1}\frac{\partial{u}}{\partial{y}}(x, y_1)dx \end{align*}
defines a harmonic conjugate for $u$ on $D(z_0;r)$ with $v(x_0, y_0) =c$.
I have done a few problems dealing with these concepts, but I'm lost as to where to start here. I've been given the hint already that it will suffice to show that $f = u+iv$ is analytic (i.e., show that the Cauchy-Riemann equations hold), but for this problem, I'm not even sure how to do this.
Any help appreciated, this is giving me a headache.