Direct integration of PDEs

What are the rules for integrating a PDE of say 2 or more variables? And where can i learn more about these ? For example, i am asked to find the solution by direct integration of

$\displaystyle u_{xyz}(x,y,z) = e^{x} + xy\;$

Now $\displaystyle \int u_{xyz}(x,y,z)\; dx = \int (e^{x} + xy)\; dx = e^{x} + (1/2)x^{2}y + g(y) + h(z) = u_{yz}$ ; And $\displaystyle \int u_{yz} dy = \int [e^{x} + (1/2)x^{2}y + g(y) + h(z)]\; dy $$\displaystyle = ye^{x} +(1/4)x^{2}y^{2} +\int g(y)\;dy +yh(z) = u_{z}; So\; that$$\displaystyle u = \int [ye^{x} +(1/4)x^{2}y^{2} +\int g(y)\;dy +yh(z)]\; dz $$\displaystyle \; = {\bf yze^{x} + (1/4)x^{2}y^{2}z + z\int g(y)\;dy + y\int h(z)\;dz + f(x)}$

Have i done this correctly ? I can't seem to find much information on integrating PDEs online. I am unsure of the arbitrary f(x),g(y) and h(z) functions, and how i have worked with them? Any help is much appreciated!