Conjugate harmonic function

[qwesd.211]

Calculate the conjugate harmonic function in the indicated domains.

$$u(x,y)=x^2-y^2+x\,\,\,\,\,\,\,\,0\leq |z|< \infty$$
I did this.

$$u_x=2x+1, \,\,u_y=-2y,\,\, u_{xx}=2,\,\,u_{yy}=-2$$
$$u_{xx}+u_{yy}=0\to 2-2=0$$
$$2x+1=v_y, \,\, -2y=-v_x$$
$$v=\int{(2x+1)dy}\to v=2xy+y+g(x)$$
$$v_x=2y+g'(x)=2y\therefore g'(x)=2y-2y\to g(x)=\int(0)dx=C$$
$$v(x,y)=2xy+y+C$$
$$f(z)=u(x,y)+iv(x,y)$$$$f(z)=(x^2-y^2+x)+i(2xy+y+C)$$
How do I use the domains?
please help :c