Sorry for the wrong forum.
I think I have done substitution correctly - extracting k^{2}cos^{2}(kr) from the expression in the first integral gives:
\int...
I'm trying to compute following integral (Wolfram doesn't give answer):
\int\sqrt{E-Bk^{2}\frac{cos^{2}(kr)}{sin^{2}(kr)}-k\frac{cos(kr)}{sin(kr)}\sqrt{D+Fk^{2}\frac{cos^{2}(kr)}{sin^{2}(kr)}}-\frac{Ak^{2}}{sin^{2}(kr)}}dr
where A,B,C,D,E,F,k are constants.
Substitution t=sin(kr) leads to...
I know how to solve this equation and how to find F in any "regular" domain, for example in real plane \mathbb{R^{2}}. Problems appear in the neighbourhood of point (0,0) in our domain, because all methods of solving this kind of equation, I know are valid only in simply connected domain.
I'm not sure, if we can use the line integral here. We are trying to show, that there doesn't exist function F such that F is exact differential, that is
\frac{ \partial F}{ \partial x}=\frac{-y}{ x^{2}+y ^{2} } and \frac{ \partial F}{ \partial y}=\frac{x}{ x^{2}+y ^{2}}
Existence of such...
I try to show, that equation
\frac{-y}{ x^{2}+y ^{2} } + \frac{x}{ x^{2}+y ^{2}}y'=0
is not exact in \mathbb{R^{2}} \setminus \{(0,0)\}.
It's obvious that I have to use the fact, that the set is not simply connected, but I don't know how to do it.