For the given u(x,y) find a v(x,y) such that f(z) = u(x,y) + iv(x,y) is holomorphic in some region of $\mathbb{C}$ I used the Cauchy Riemann equations and got two different functions v(x,y) by integrating.  Should I just add them up?  
My u(x,y) is $\frac{x}{x^2 + y^2}$
My partial wrt x is $\frac{y^2 - x^2}{x^2 + y^2}$ and my partial wrt y is $\frac{-2xy}{x^2 + y^2}$.

$u(x,y)$ partial wrt $x$ is $\frac{y^{2}-x^{2}}{(x^2+y^2)^2}$, partial wrt $y$ is
$\frac{-2xy}{(x^2+y^2)^2}$.
By the rule of Cauchy-Riemann Equations,we have
$v(x,y)$ partial wrt $y$ is $\frac{y^{2}-x^{2}}{(x^2+y^2)^2}$, partial wrt $x$ is
$\frac{-2xy}{(x^2+y^2)^2}$.
Then we can let $v(x,y)=\frac{-y}{x^2+y^2}$.
