All Questions
Tagged with integral or integration
171 questions
84
votes
1
answer
7k
views
A hard integral identity on MathSE
The following identity on MathSE
$$\int_0^{1}\arctan\left(\frac{\mathrm{arctanh}\ x-\arctan{x}}{\pi+\mathrm{arctanh}\ x-\arctan{x}}\right)\frac{dx}{x}=\frac{\pi}{8}\log\frac{\pi^2}{8}$$
seems to be ...
- 3,337
43
votes
1
answer
2k
views
Is $\int_0^\infty{dx\over x^{x^{x^x}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^x}}}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^{x^{x^x}}}}}}}<\cdots$ true?
On MSE I asked whether each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ was less than $2$ and received answers on bounding ...
- 1,459
43
votes
3
answers
5k
views
Is there a systematic method for differentiating under the integral sign?
This MO question by Tim Gowers reminded me of a question I've wondered about for some time. In the delightful book Surely You're Joking, Mr. Feynman!, Feynman praises the technique of differentiating ...
- 82.7k
42
votes
0
answers
2k
views
Are we better in computing integrals than mathematicians of 19th century?
When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ...
33
votes
3
answers
4k
views
Can we simplify $\int_{0}^{\infty}\frac{{\sin}^px}{x^q}dx$?
We know the followings :
$$\int_{0}^{\infty}\frac{{\sin}x}{x}dx=\int_{0}^{\infty}\frac{{\sin}^2x}{x^2}dx=\frac{\pi}{2},\int_{0}^{\infty}\frac{{\sin}^3x}{x^3}dx=\frac{3\pi}{8}.$$
Also, we can get
$$\...
- 4,757
32
votes
3
answers
3k
views
What are the obstructions for a Henstock-Kurzweil integral in more than one dimension?
I have recently come across the book The Kurzweil-Henstock Integral and its Differentials by Solomon Leader, in which, if I understand correctly, the HK integration process is modified in a way that ...
- 573
30
votes
4
answers
2k
views
If $X$ and $Y$ independent and identically distributed, then $E(|X-Y|)\leq E(|X+Y|)$. Are other proofs of this known?
I know a proof of the theorem that if $X$ and $Y$ independent and identically distributed, then $E(|X-Y|)\leq E(|X+Y|)$. The proof uses an integral representation of the absolute value,
$$\int_0^\...
- 409
29
votes
2
answers
2k
views
What theorem constructs an initial object for this category? (Formerly "Integrability by abstract nonsense")
Tom Leinster has a note here about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than ...
- 12.1k
26
votes
2
answers
12k
views
About the definition of Borel and Radon measures
I am trying to understand the notion of Radon measure, but I am a little bit lost with the different conventions used in the litterature.
More precisely, I have a doubt about the very definition of ...
- 281
25
votes
2
answers
2k
views
Interesting integral
Numerical evidence shows the validity of the following identity
$$\int\limits_0^z\frac{xdx}{\sin{x}\sqrt{\sin^2{z}-\sin^2{x}}}=\frac{\pi}{4\sin{z}}\ln{\frac{1+\sin{z}}{1-\sin{z}}},\tag{1}$$
if $0< ...
- 16.5k
24
votes
2
answers
1k
views
Is $\iiint_{[0, 1]^3} \lvert f(x)+f(y)+f(z)\rvert\, dx\, dy\, dz \ge \int_0^1 \lvert f(x)\rvert\, dx$?
$\newcommand\abs[1]{\lvert#1\rvert}\newcommand\Abs[1]{\left\lvert#1\right\rvert}$Question: Let $f(x)$ be a continuous real-valued function defined on the interval $[0, 1]$. Does the following ...
- 4,280
23
votes
4
answers
5k
views
Is $\ x\! \cdot\!\tan(x)\ $ integrable in elementary functions?
I'm teaching Calculus and my students asked me to calculate the integral of $\ x\! \cdot\!\tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be ...
- 1,437
19
votes
5
answers
18k
views
Visualization of Riemann–Stieltjes Integrals
The Riemann–Stieltjes integral $\int_a^b f(x)\,dg(x)$ is a generalization of the Riemann integral. It is e.g. heavily used as a starting point for stochastic integration. The approximating Riemann–...
- 5,935
18
votes
2
answers
1k
views
An interesting triple integral
What is the value of this triple integral $$\int\limits_0^{2\pi}\int\limits_0^{2\pi}\int\limits_0^{2\pi}|\cos x+\cos y+\cos z|\ dx\ dy\ dz?$$ It has to do with some Schwarz lemma.
- 719
17
votes
3
answers
1k
views
Decoupling a double integral
I came across this question while making some calculations.
QUESTION. Can you find some transformation to "decouple" the double integral as follows?
$$\int_0^{\frac{\pi}2}\int_0^{\frac{\pi}...
- 43.2k
17
votes
2
answers
850
views
Non-negative polynomials on $[0,1]$ with small integral
Let $P_n$ be the set of degree $n$ polynomials that pass through $(0,1)$ and $(1,1)$ and are non-negative on the interval $[0,1]$ (but may be negative elsewhere).
Let $a_n = \min_{p\in P_n} \int_0^1 ...
- 799
16
votes
2
answers
819
views
Numerical integration using interval arithmetic, nowadays
This is an update to my question Rigorous numerical integration from three years ago.
Is there now a package for rigorous numerical integration that uses interval arithmetic and has access to a well-...
- 20.2k
15
votes
3
answers
2k
views
Riesz's representation theorem for non-locally compact spaces
Every version of Riesz's representation theorem (the one expressing linear functionals as integrals) that I have found so far assumes that the underlying topological space is locally-compact. (For ...
- 5,407
14
votes
2
answers
2k
views
Computing Gauss Legendre quadrature for large $N$
I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscissas and weights $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it,...
- 3,574
14
votes
2
answers
807
views
Integral of power of binomials equal to sum of power of binomials?
Inspired by this MO question about integrating binomial coefficients and the answers, I was wondering whether integrating powers of binomial coefficients also relates to the respective sums. And ...
- 4,014
13
votes
2
answers
743
views
How to prove that $\int _0^\infty\frac{\text{arcsinh}^nx}{x^m}dx$ is a rational combination of zeta values?
For $n\ge m\ge 2$, define $$I(n,m):= \int _0^\infty\dfrac{\text{arcsinh}^nx}{x^m}dx$$ Computer algebra systems say that the indefinite integral can be expressed in terms of polylog functions (of ...
- 13.4k
13
votes
3
answers
823
views
Is there a closed form of $\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$?
For naturals $n\ge m$, define
$$I(n,m):=\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$$
with $\text{arcsinh}\ x=\ln(x+\sqrt{1+x^2} )$, so $\text{arcsinh} \frac12=\ln \frac{\sqrt{5}+1}2 $.
Is it ...
- 13.4k
13
votes
1
answer
761
views
If $(a,b,c)$ are the sides of a triangle, then the probability $P(ax + by \ge c) = \frac{4}{\pi^2}\chi_2(x) + \frac{4}{\pi^2}\chi_2(y)$
Posting this question in MO since it is unanswered in MSE
Let $(a,b,c)$ be the side of a triangle. In its most general linear form, the triangle inequality can be expressed as: Does $ax + by \ge c$ ...
- 5,470
12
votes
1
answer
2k
views
Naive definition of surface area doesn't work?
A first stab at a definition of surface area might go like this:
Let S be a surface. Select finitely many points from S and make a bunch of triangles having these points as vertexes. Add up the ...
- 12.1k
12
votes
2
answers
1k
views
Counterexamples to differentiation under integral sign, revisited
Let $f\colon\mathbb R^2\to\mathbb R$ be a measurable function such that
\begin{equation*}
F(t):=\int_{\mathbb R}dx\,f(t,x)
\end{equation*}
exists and is finite for all real $t$. Suppose that
\...
- 128k
11
votes
2
answers
603
views
Function orthogonal to powers of $1/\left(1+x^2\right)$
Does there exist any continuous function $f:\mathbb{R}\to\mathbb{R}$, $f(x)/(1+x^2)\in L^1(\mathbb R)$, such that $f(0)=1$ and
$$\int_{-\infty}^{\infty}\frac{f(x)}{\left(1+x^2\right)^p}dx=0$$ for ...
- 537
10
votes
2
answers
1k
views
Can the integration of integrable sections of a measurable function of two variables ever result in a non-measurable function?
I spent some time searching MathOverflow for a problem that would resemble the one given below, but it turned out to be a rather futile endeavor. I was led to this problem in my attempts to construct ...
- 1,396
10
votes
1
answer
385
views
When is this multiple integral finite?
Consider the following integral:
$$
I_k(\alpha)=\int_{[0,1]^k}|x_1-x_2|^{\alpha}|x_2-x_3|^{\alpha}\ldots|x_{k-1}-x_k|^{\alpha}|x_k-x_1|^{\alpha}d\mathbf{x}.
$$
where $k=2,3,4,\ldots$
The question is ...
- 87
9
votes
3
answers
3k
views
What is the Dunford Integral and why is it useful?
Wikipedia http://en.wikipedia.org/wiki/Pettis_integral defines the Pettis Integral for Banach space valued functions wrt to some measure space by duality.
It calls the Pettis & Bochner integral ...
- 2,083
9
votes
1
answer
451
views
Improper integral $\int_0^1 \frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx$ with $-a$ and $b$ positive
Is the following function real analytic in $t>0$:
$$F(t)=\int_0^1\frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx,$$
where $-a$ and $b$ are positive, and $c\not=a$?
...
- 353
9
votes
2
answers
646
views
Rain droplets falling on a table
Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
- 191
9
votes
2
answers
622
views
Integral inequality: Prove $\int_0^1 f\int_0^1 1/f \leq 1$ for a certain function $f$
Let $g$ be a piecewise smooth, zero average, function over $[0,1]$ such that $\min g^2>0$. I would like to show that
$$
\int_0^1 g\sqrt{1-r/g^2}\int_0^1 \frac{1}{g\sqrt{1-r/g^2}} \leq 1
$$
for all $...
- 264
8
votes
2
answers
948
views
Interesting triple integral
Some time ago I stumbled on an alleged identity
$$\int\limits_0^\infty \frac{dx}{x} \int\limits_0^x \frac{dy}{y}
\int\limits_0^y \frac{dz}{z} [\sin{x}+\sin{(x-y)}-\sin{(x-z)}-\sin{(x-y+z)}]=
-\frac{\...
- 16.5k
8
votes
2
answers
1k
views
A curious family of integrals that give $\pi$
I have noticed experimentally that:
$$\int_0^1 \frac{\color{red}{x}}{x^4+2x^3+2x^2-2x+1} \,dx=\color{blue}{\frac{\pi}{8}},\tag{1}$$
$$\int_0^1 \frac{\color{red}{1-x^2}}{x^4+2x^3+2x^2-2x+1} \, dx=\...
8
votes
2
answers
828
views
Convergence of an oscillatory integral
Given $f\in H^1(\mathbb{R}^3)$ and $t>0$, consider the following integral:
$$I_f(t):=\int_{\mathbb{R}^3}\int_0^{+\infty}e^{-s+i\frac{(s+|x|)^2}{t}}f(x)dsdx$$
I need to show that $I_f(t)$ is finite, ...
- 943
7
votes
4
answers
2k
views
Integrating a barycentric monomial over a simplex
Are there standard formulas for the integral over a simplex of a monomial in the barycentric coordinates? Can someone supply a reference? I think I have seen such formulas, but I am unable to find ...
- 301
7
votes
3
answers
1k
views
Action Integral
In the theory of action-angle variables, you wind up having to solve integrals with a characteristic square-root behavior near the endpoints to express the action in terms of the orbital quantities. ...
- 73
7
votes
4
answers
1k
views
Integral of product of gaussian CDF and PDF
Looking for an analytic solution to the integral below:
$$
\int_{-\infty}^\infty \Phi\left(\frac{x - a}{\tau}\right) \phi\left(\frac{x - b}{\sigma}\right)dx
$$
where $\Phi(\cdot)$ and $\phi(\cdot)$ ...
- 73
7
votes
1
answer
552
views
Dominated convergence 2.0?
During my research, I came across the following question.
Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging pointwise to $g \in L^1([0,1])$. Assume that:
$\forall n\in\mathbb N, f_n''<h$, ...
- 4,074
7
votes
2
answers
464
views
Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace
Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
- 12.5k
7
votes
5
answers
1k
views
Generalizations of the Euler–Maclaurin Summation Formula
I'm using the Euler–Maclaurin formula in a research project I'm working on. While brilliant, the elementary proof found in Apostol - An Elementary View of Euler's Summation Formula does not give me ...
- 3,574
7
votes
1
answer
552
views
Integrating the multinomial over a hypercube
I have come across an integral of the form
$$\int_{b}^{a}\cdots\int_{b}^{a} \left( \sum_{i=1}^{n}x_i\right)^mdx_1d x_2\dots dx_n.$$
I have a solution that makes use of the partition function, but I ...
- 1,026
6
votes
3
answers
713
views
Expected absolute value of the average of two points from the disc
Looking at Average distance of the mean of n random complex numbers in a unit disc, I tried to figure out
what is the expected absolute value $|\frac{z_1 + z_2}{2}|$ of two numbers $z_1, z_2\in\...
- 10.7k
6
votes
3
answers
8k
views
Proof without words for surface area of a sphere [closed]
I love the book Proofs Without Words by Roger B. Nelsen. One of the proofs I liked the most was this: Area under one arch of a cycloid is 3 times the area of the wheel that traces it. You break the ...
- 106
6
votes
0
answers
292
views
A kind of reflection formula for the logarithmic derivative of the zeta function
So I was messing around with Bernoulli numbers and values of $\zeta'$ at integers $-$ and suddenly I came about a non trivial identity which can be written in terms of the logarithmic derivative of ...
- 13.4k
6
votes
1
answer
312
views
Making sense out of intertwining operators defined by a vector valued integral
Let $G$ be the rational points of a connected, reductive group over a $p$-adic field $F$. Let $S$ be a maximal split torus of $G$ with $\Delta$ a set of simple roots corresponding to a minimal ...
- 6,180
6
votes
0
answers
1k
views
Evaluating $\iint_{\mathbb{R}\times \mathbb{R}^{+}}e^{-|w-c_{1}|^2-|u-c_{2}|^2}\frac{1}{w_{1}+iw_{2}-u_{1}-iu_{2}}dw_{1}dw_{2}du_{1}du_{2}$
For $c_{1},c_{2}\in \mathbb{H}:=\{Im(z)>0\}$ I want to compute the following integral or prove it doesn't exist:
$$\int_{\mathbb{R}\times \mathbb{R}^{+}}\int_{\mathbb{R}\times \mathbb{R}^{+}}e^{-|...
- 5,474
6
votes
1
answer
400
views
Densities, pseudoforms, absolute differential forms and measures, differential forms, etc
Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here.
Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
- 831
6
votes
1
answer
1k
views
Fubini's theorem on arbitrary foliations
In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$
Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of $...
- 622
6
votes
2
answers
635
views
Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?
PROBLEM. Let $\theta(t)$ and $\phi(t)$ be two real analytic non-constant functions $[0,2\pi]\rightarrow \mathbb{R}$. I am trying to prove the following claim
If the integral
$$
\int_0^{2\pi} e^{i\...
- 405