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Results tagged with integration
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user 13972
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, ...
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 …
- 108k
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
$$
V = f(x)\,\fra …
- 108k
5
votes
Accepted
Solution of this differential equation
Yes, this can be integrated explicitly. First, notice that, since $m\not=0$, we can write $\alpha(t) = 2m\bigl(x(t)+iy(t)\bigr)$, in which case, the given equation becomes
$$
\dot x + i\,\dot y = -4( …
- 108k
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 …
- 108k
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 …
- 108k
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) …
- 108k