Answer
$f(x, y, z) = \frac{1}{2} \ln(x^2+y^2) + z + K$
Work Step by Step
\[
\mathbf{F}(x, y, z) = \frac{x}{x^2+y^2}\mathbf{i} + \frac{y}{x^2+y^2}\mathbf{j} + 1\mathbf{k}
\]
\[
\nabla \times \mathbf{F} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\
\dfrac{x}{x^2+y^2} & \dfrac{y}{x^2+y^2} & 1
\end{vmatrix}
\]
\[
= \left(\frac{\partial}{\partial y}1 - \frac{\partial}{\partial z}\frac{y}{x^2+y^2}\right)\mathbf{i}
- \left(\frac{\partial}{\partial x}1 - \frac{\partial}{\partial z}\frac{x}{x^2+y^2}\right)\mathbf{j}
+ \left(\frac{\partial}{\partial x}\frac{y}{x^2+y^2} - \frac{\partial}{\partial y}\frac{x}{x^2+y^2}\right)\mathbf{k}
\]
\[
= 0\mathbf{i} - 0\mathbf{j} + 0\mathbf{k}
\]
\[
\therefore \mathbf{F} \text{ is conservative in any domain excluding } (0,0,z)
\]
\[
f(x, y, z) = \int f_x\,dx = \int \frac{x}{x^2+y^2}\,dx = \frac{1}{2} \ln(x^2+y^2) + g(y, z)+K1
\]
\[
f(x, y, z) = \int f_y\,dy = \int \frac{y}{x^2+y^2}\,dy = \frac{1}{2} \ln(x^2+y^2) + h(x, z) +K2
\]
\[
f(x, y, z) = \int f_z\,dz = \int 1\,dz = z + k(x, y)+K3
\]
\[
\text{Combine: } f(x, y, z) = \frac{1}{2} \ln(x^2+y^2) + z + K
\]
\[
\boxed{f(x, y, z) = \frac{1}{2} \ln(x^2+y^2) + z + K}
\]
