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

Now ; And

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!