These cropped up in one of my calculations. I have forgotten if they are even possible to integrate and I can't find the reference, so I'd appreciate a hand. (I will note that at least one of these looks suspiciously like an elliptic integral.)
We may take b > a > 0 WLOG.
Any help/thoughts would be appreciated.
-Dan
all three integrals are linear combinations of elliptic integrals of the first and the second kind.
i'll give you the solution for the first integral. the other two have the same idea:
let and
it's easy to see that which we'll call it now we have the following:
thus by we have:
now we find and in terms of elliptic integrals of the second and first kind respectively: put then:
similarly we have: thus by we will have:
.................................................. .................................................. .
Remark: you can also find in the form of an infinite series:
since we'll get: