A Particular Integrals of PDEs

[vgarg]

TL;DR Summary

Partial integration

Can someone please explain the steps for the integrations in red circles on the attached page? This a page from Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering 2006 edition.
Thank you.

 

[PeroK]

TL;DR Summary: Partial integration

Can someone please explain the steps for the integrations in red circles on the attached page? This a page from Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering 2006 edition.
Thank you.

If you have a function of a single variable whose derivative is zero, then that function must be a constant. If you have a function of two variables whose derivative is zero with respect to one of the variables, then it must be a function only of the other variable.

 

[PeroK]

PS you should try to find a function of two variables whose mixed second partial derivative is zero. In general, I'd you don't understand something like this, trying to find a counterexample or otherwise prove it's wrong is is good way to understand why it's correct.