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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, ...
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
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
$$
\dot x + i\,\dot y = -4( …
- 108k
1
vote
Class of analytically-integrable divergence-free vector fields?
You probably need to specify more conditions, as the ones you give are too loose to be interesting. For example, consider the divergence-free vector field in the $xy$-plane given by
$$
V = f(x)\,\fra …
- 108k
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
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
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