Chapter 2.4, Problem 36E

To determine

To verify:

The function $F\left(x,y\right)$ defined by $F\left(x,y\right)={\displaystyle \underset{x_0}{\overset{x}{\int }}M\left(t,y\right)dt}+g\left(y\right)$ and $g\text{'}\left(y\right)=N\left(x,y\right)-\frac{\partial }{\partial y}{\displaystyle \underset{x_0}{\overset{x}{\int }}M\left(t,y\right)dt}$ satisfies the condition $\frac{\partial F}{\partial x}=M,\frac{\partial F}{\partial y}=N$.