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Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

162 votes
Accepted

Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?

represents a differential that has two poles of order $2$ (over the two points where $t=\infty$) and two poles of order $1$ (over the two points where $t=0$), an application of Liouville's Theorem (on integration
Robert Bryant's user avatar
108 votes

Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?

I'm adding a separate answer for the general question that the OP asked, which settles the question in the negative for all $n>2$ (and gives an alternate proof for $n=3$ to the one I gave above). Rec …
Robert Bryant's user avatar
36 votes
Accepted

Interesting integral

Actually, I now think that the easiest method is to do this: Write $k=\sin z$, so that $|k|<1$, and make the substitution $x = \arcsin(k\sin\theta)$, where $0\le \theta\le \frac\pi2$. The integral b …
Robert Bryant's user avatar
5 votes
Accepted

Solution of this differential equation

Yes, this can be integrated explicitly. First, notice that, since $m\not=0$, we can write $\alpha(t) = 2m\bigl(x(t)+iy(t)\bigr)$, in which case, the given equation becomes $$ \dot x + i\,\dot y = -4( …
Robert Bryant's user avatar
4 votes

Volume of submanifold as integral of delta-function

One should distinguish between the volume of the submanifold (a number that might be infinite) and the volume form, an exterior differential form $\omega$ of degree $n{-}m$ on the (presumed regular) …
Robert Bryant's user avatar
1 vote

Class of analytically-integrable divergence-free vector fields?

You probably need to specify more conditions, as the ones you give are too loose to be interesting. For example, consider the divergence-free vector field in the $xy$-plane given by $$ V = f(x)\,\fra …
Robert Bryant's user avatar