Find the general integral of the equation (∂ z)/(∂ x) = x + y

Find the general integral of the equation \[ \frac{\partial z}{\partial x} = x + y. \]

Explanation of Method

The equation gives $z$ directly by integrating with respect to $x$, treating $y$ as a parameter; the constant of integration becomes an arbitrary function of $y$.

Step 1 · Integration

\[ z = \int (x + y)\,dx = \frac{x^2}{2} + xy + \varphi(y), \] where $\varphi(y)$ is an arbitrary differentiable function of $y$ alone, playing the role of the constant of integration.

Verification

Differentiate with respect to $x$: \[ \frac{\partial z}{\partial x} = x + y. \checkmark \]

Result

The general integral is:

\[ \boxed{z = \frac{x^2}{2} + xy + \varphi(y)} \]