The integration tag has no wiki summary.

**76**

votes

**5**answers

6k views

### Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference:
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
where the ...

**72**

votes

**17**answers

15k views

### Why is differentiating mechanics and integration art?

It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by ...

**54**

votes

**1**answer

3k views

### A hard integral identity on MATH.SE

The following identity on MATH.SE
$$\int_0^{1}\arctan\left(\frac{\mathrm{arctanh}\ x-\arctan{x}}{\pi+\mathrm{arctanh}\ x-\arctan{x}}\right)\frac{dx}{x}=\frac{\pi}{8}\log\frac{\pi^2}{8}$$
seems to be ...

**49**

votes

**2**answers

2k views

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

I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here:
Is it possible to express ...

**48**

votes

**8**answers

6k views

### Why is Lebesgue integration taught using positive and negative parts of functions?

Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only ...

**30**

votes

**3**answers

2k 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 ...

**23**

votes

**3**answers

3k views

### What is the standard notation for a multiplicative integral?

If $f: [a,b] \to V$ is a (nice) function taking values in a vector space, one can define the definite integral $\int_a^b f(t)\ dt \in V$ as the limit of Riemann sums $\sum_{i=1}^n f(t_i^*) dt_i$, or ...

**19**

votes

**2**answers

1k 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 ...

**19**

votes

**2**answers

1k 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 ...

**17**

votes

**6**answers

2k views

### Why not evaluate integrals using ODE-solvers?

Hello!
I have a question about the relationship between numerical integration and the solution of ordinary differential equations (ODE). Suppose I want to evaluate the integral
$I(x) = \int_{0}^{x} ...

**17**

votes

**1**answer

525 views

### An NP-hard $n$ fold integral

We are given rational numbers $[c_1, c_2, \ldots, c_n]$ and $v$ from the interval $[0,1]$.
Consider the $n$-fold integral
$$
J = \int_{\theta_1 \in I_1, \theta_2 \in I_2 \ldots, \theta_n \in I_n} ...

**16**

votes

**2**answers

865 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
...

**16**

votes

**3**answers

2k views

### Weak and Strong Integration of vector-valued functions

This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference:
Suppose $f:X\to E$ is a continuous function from a compact spaces (endowed ...

**15**

votes

**4**answers

2k views

### Is $x \, \tan(x)$ integrable in elementary functions?

I'm teaching Calculus and my students asked me to calculate the integral of $x \, \tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be presented in ...

**14**

votes

**2**answers

492 views

### Evaluation of an $n$-dimensional integral

I asked the same question on math.se but got no answer there. Since it pertains to my current research, I decided to ask here:
Let $n\in 2\mathbb{N}$ be an even number. I want to evaluate
$$I_n
:=
...

**14**

votes

**1**answer

949 views

### A mass spring model for hair simulation

A strand of hair is represented by a set of particles connected by springs.
The velocity for a particular particle is calculated implicitly using the following formula:
...

**12**

votes

**4**answers

2k views

### Integrals from a non-analytic point of view

I've mentioned before that I'm using this forum to expand my knowledge on things I know very little about. I've learnt integrals like everyone else: there is the Riemann integral, then the Lebesgue ...

**12**

votes

**5**answers

1k views

### An algebra of “integrals”

When discussing divergent integrals with people, I got curious about the following:
Is there an $\mathbb{R}$-algebra $A$ together with a map (could be defined on just a subspace)
$$\int_0^{\infty}: ...

**12**

votes

**1**answer

647 views

### Integration in the surreal numbers

In the appendix to ONAG (2nd edition), Conway points that the definition of integration (using Riemann sums as left and right options) gives the "wrong" answer : $\int_0^\omega \exp(t)\thinspace ...

**11**

votes

**5**answers

2k views

### Why do I need densities in order to integrate on a non-orientable manifold?

Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form ...

**11**

votes

**4**answers

6k views

### Visualization of Riemannâ€“Stieltjes Integrals

The Riemannâ€“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 ...

**11**

votes

**2**answers

852 views

### Is there a notion of integration over the algebraic numbers?

For reasons which are hard to articulate (due to they not being very clear in my mind), but having to do with the eprint From Matrix Models and quantum fields to Hurwitz space and the absolute Galois ...

**11**

votes

**3**answers

831 views

### Applications of visual calculus

Mamikon's visual calculus (see Mamikon, Tom Apostol, Wikipedia) is a very beautiful and surprisingly efficient tool.
The basis is
Mamikon's theorem. The area of a tangent sweep is equal to the ...

**10**

votes

**3**answers

2k views

### Counterexamples to differentiation under integral sign?

I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both ...

**10**

votes

**6**answers

3k views

### Numerical integration over 2D disk

I have a real-valued function $f$ on the unit disk $D$ that is fairly well behaved (real-analytic everywhere) and would like to find the integral $\int_D f(x,y)dxdy$ numerically. After much searching, ...

**10**

votes

**1**answer

591 views

### Is the pairing between contours and functions perfect (modulo the kernel given by Stokes' theorem)?

Let $s: \mathbb C^n \to \mathbb C$ be a homogeneous degree-$d$ polynomial which is nonsingular (in the sense that the hypersurface it defines in $\mathbb{CP}^{n-1}$ is smooth; equivalently the ...

**10**

votes

**1**answer

327 views

### Modern version of an inequality of R. M. Gabriel for contour integrals

I am currently reading the 1998 article Dynamics of the Binary Euclidean Algorithm:
Functional Analysis and Operators by Brigitte VallÃ©e, which cites a 1928 article by R. M. Gabriel for the following ...

**9**

votes

**2**answers

698 views

### Borel sets preserved under open maps?

Given open map f: $R^n$ to $R^n$ such that each open set $U\in R^n$, $f(U)$ is also open. Are Borel sets in $R^n$ preserved under f?
Motivation: Pre-image of Borel sets under continuous map is a ...

**9**

votes

**0**answers

185 views

### A multiple integral

Let us consider the multiple integral
$$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots
\int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots
\cos ...

**8**

votes

**4**answers

2k views

### An integral that somehow equals pi^2/6 and involves dilogarithms?

I am attempting to show that
$$ \sum_{k \ge 1}^\infty {k^2 x^k \over (1+x^k)^2} \sim (1-x)^{-3} {\pi^2 \over 6} $$
as $x$ approaches 1 from below. The sum can be approximated by the integral
$$ ...

**8**

votes

**4**answers

419 views

### Computing $\int_0^T e^{itA}Be^{-itA} dt$ without an infinite series

I'm hoping to compute the following integral: $\int_0^T e^{itA}Be^{-itA} dt$ where $iA, iB$ are traceless anti-Hermitian matrices (i.e. $\mathfrak{su}(n)$). I have found the following form for the ...

**8**

votes

**2**answers

583 views

### An identity involving an infinite integral with a sinh in the denominator

I recently encountered the rather appealing looking integral, which appears in the theory of random matrices :
$$\int_{-\infty}^{\infty}\prod_{j=1}^{p-1}(j^{2}+z^{2})\frac{zdz}{\mathrm{sinh}(2\pi z)} ...

**8**

votes

**2**answers

327 views

### $\mathrm{Bessel}^3$ Integral

I'm trying to calculate the following integral:
$\int_0^\infty \mathrm{BesselJ}[l_0,k_0r] \cdot \mathrm{BesselJ}[l_1,k_1r] \cdot \mathrm{BesselJ}[l_0-l_1,kr] \cdot r\,dr$
($\mathrm{BesselJ}[n,x]$ is ...

**8**

votes

**2**answers

482 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 ...

**8**

votes

**1**answer

240 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 ...

**8**

votes

**1**answer

428 views

### Certain compact subset of $L_1$

Let $(\Omega,\Sigma, \mu)$ be a probability measure and $X$ a Banach space. I am interested in subsets $F\subseteq L_\infty (\mu,X)$ that satisfy these two compactness conditions:
$F$ is a ...

**8**

votes

**1**answer

1k views

### About the surface area vs. volume of polytopes

Given a convex body $K\in\mathbb{R}^n$, represented by a set of linear inequalities (intersection of halfspaces), I am interested in understanding how much of its volume can be close to its perimeter ...

**8**

votes

**2**answers

297 views

### On the convexity of certain integrals involving Bessel functions

Let $n\geq 0$ be an integer and let $J_n=J_n(r)$ denote the usual Bessel function (of the first kind) of order $n$ i.e. one of the solutions to Bessel's differential equation
...

**7**

votes

**5**answers

740 views

### $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$

I'm trying to solve the integral
$\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$,
where $s$, $r$ and $m$>1 are positive integers.
My question is whether a closed form ...

**7**

votes

**3**answers

1k views

### Rationale for Hadamard's finite part of a divergent integral

(Note: I asked this question a few days ago on math.stackexchange but didn't get any responses. I've therefore decided to post it here instead.)
I have a problem justifying throwing away the ...

**7**

votes

**3**answers

745 views

### Expectation of a simple function of multivariate gaussians iid rvs

I would like to compute analytically the following expected value:
$$ E\left( \frac{X_i^2}{\sum_j \lambda_j^2 X_j^2}\right) $$
where the $X_i \approx N(0,1)$ are iid.
It seems to be an elementary ...

**7**

votes

**3**answers

751 views

### Nontrivial trivial integrals

I posted this question to stackexchange and after 24 hours it's got five votes and no answers, so let's see if mathoverflow can say more than that.
Consider two propositions in geometry:
...

**7**

votes

**3**answers

258 views

### Multiprecision numerical evaluation of integral: Sage vs. PARI/GP vs. mpmath

I am trying to compute thousands of integrals of the below type, that comes up in a conformal mapping problem, to as many accurate digits as possible (preferably 50+):
$$
\int_{-1}^1\textrm{d}t ...

**7**

votes

**2**answers

720 views

### Contour integration problem from probability

Can integrals of the form
$$
\int_{-\infty}^{\infty}{\exp\left(-\left[x - c\right]^{2}\right) \over 1 + x^{2}}\, {\rm d}x
$$
be computed in closed form using contour integration (or any other ...

**7**

votes

**1**answer

810 views

### Universally measurable sets and weak topology

After I posted this question, a couple of months ago, and got from MO-users several
good hints, I think i'm ready, after some study, to ask another related question (or rather, to focus on the main ...

**7**

votes

**2**answers

277 views

### Integration on Compact Semirings

I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is ...

**6**

votes

**2**answers

2k 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 ...

**6**

votes

**2**answers

146 views

### Numerical integration of legendre polynomials

I hope that numerical questions are also permitted here.
I want to expand a smooth functions $f \in C^{\infty}$in terms of Legendre polynomials. Thus I need to calculate integrals of the form ...

**6**

votes

**1**answer

230 views

### “Values” of divergent integrals

Are there existing theories of integration in which $I_0 = \int_0^{\infty} dx$ and $I_1 = \int_0^{\infty} x \ dx$ are well-defined infinite elements in a non-archimedean extension of the reals? I can ...

**6**

votes

**0**answers

295 views

### Minkowski's Inequality for Integrals in Orlicz spaces

EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces.
Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, ...