Your original question had $1/\root3\of{2x^4}$. Have you changed the question? Your answer is correct if the question is $\root3\of{2x^4}$ ... not in the denominator.
@Andrii: When you write $dR$, you really mean $d\vec R$. And using $R$ for the radius of the circle is totally confusing. Now what does $1/\vec R$ mean?
A line integral (or you're calling it a curve integral) is ordinarily for the work done by a vector field along a path, so one writes $\int_C \vec F\cdot d\vec R$.
(I've only taught this stuff for more than 45 years and written a textbook on it.)
Then write everything with exponents again and apply the power rule.
So, first of all, @Andrii, the notation in the problem is sloppy, but they're integrating with respect to arclength around the circle. But the function is very complicated.
@Andrii: The function in question involves distance from a point on the curve to a fixed point $(x,y)$ and they want you to compute the thing as a function of $(x,y)$.
@TedShifrin you might be able to do it by hand by appealing to the identity $$\sum_{n=0}^\infty T_n(x)\frac{x^n}{n!} = \ln \frac{1}{\sqrt{1-2tx+t^2}}$$ where $T_n(x)$ is the $n$th Chebyshev polynomial of the second kind
@TedShifrin So I was doing this entire thing wrong the whole time, huh. I get that I need to do a lot of reading on the mean value property then, thanks, for now at least
There's lots of Green's Theorem stuff. But I don't see Green's formulas for harmonic functions. The other questions are reasonably standard applications of Green's Theorem.
@Alessandro: In the US school has already started lots of places.
@TedShifrin Here we start August 27th. But the final oral exam for those upgrading their math from B to A level may be as late as the 29th (so it seems like some people forgot to actually check the dates when they arranged the exams)
@Kasmir: I was supposed to go on a trip to Stanford, Berkeley right now to see friends ... but there are fires and bad smoke in northern CA so I canceled.
@TedShifrin yeah I believe he didn't ever mention harmonic functions. Actually, I've made quite a bit of progress brute forcing this particular problem and he didn't point me out that this is a wrong approach entirely, which is why I was so confused
well, @Andrii, to apply Green's Theorem you first have to write the $u ds$ integral as an appropriate $P\,dx + Q\,dy$ integral and then you can use Green's Theorem. I honestly do not know what he's taught you. I know what I teach my students :P
@AlessandroCodenotti i think to say that a lot of the nice properties of complex stuff are really coming from ellipticity isnt just a matter of perspective but a substantive point
