I'm completely stumped on an integration problem in my work which, I suspect might be easy, but I have no clue as to how to approach it. It goes like this:
Can I simplify $\int y(u,x)dx$, where $u(x,y) = \frac{\partial}{\partial x}v(x,y)$ ?
I feel like the derivative and antiderivative should cancel in places, but I don't know how to handle the inverse relationship between $u(x,y)$ and $y(x,u)$.
Any advice would be much appreciated!
Hugh
EDIT: After originally attempting to post the issue in a more abstract manner, here's the exact situation I am dealing with:
$\int y(u,x)^2dx$, where $u(x,y) = \frac{\partial}{\partial x} v(x,y)$ and $v(x,y) = ax\left(\frac{1}{(y^2+x^2)^{1/2}} + \frac{b}{(y^2+x^2)^{3/2}}\right)$.
The issue is that, while I can carry out the derivation to find $u(x,y)$, I can't isolate $y^2$ in order to integrate it. Hence, I'm wondering if there's a way to avoid doing the derivation/integration altogether.