While I was solving some exercises as training for a test I'm gonna have, I've noticed that some integrals that I have to solve, I don't know how to do them, and I have no explanation on my papers and we didn't talk about them on my classes. I'm just assuming they are not gonna appear, but, nonetheless I would like to know how to solve them!
Having: $f(x,y) = \frac{10}{x^2-y^2}$, how can I solve $\int f(x,y) dx$?
I did the follow: $\frac{10}{x^2-y^2}$ = $\frac{A}{x-y} + \frac{B}{x+y}$, but then I'll have $x(A+B) + y(A-B) = 10$, and from now on I don't know how to solve the exercise. I tried to give a random value to $y$, since I'm solving in order do $x$, but it didn't work as well.
Thank you very much for the help!
Note that your $y$ is a "constant" and hence $A$ and $B$ are allowed to depend on $y$ (but not on $x$). From the equation $$(A+B)x + y(A-B) = 10,$$ the left-hand side must have coefficient $0$ for the $x$, since the right-hand side is a constant with respect to $x$. Hence $$A+B = 0.$$ Also, by substituting $x = 0$, you should see that $$A-B = \frac{10}{y},$$ if $y\ne 0$. Can you now solve for $A$ and $B$ (remember that they will be functions of $y$ rather than just numbers) and find the integral for $y\ne 0$?
(You should separately solve the integral in the case $y=0$. This is an easy integral.)