# Questions tagged [integration]

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, ...

981
questions

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### Elementary calculus estimate or not?

Does there exist a constant $C>0$ such that for all $f \in H^3(\mathbb R)$
$$\int_{\mathbb R} \vert x f''(x) \vert^2 \ dx \le C \int_{\mathbb R} \vert f'''(x) \vert^2 + \vert x^3f(x) \vert^2 + \...

**14**

votes

**1**answer

519 views

### Almost all non-negative real numbers have only finitely many multiple lies in a measurable set with finite measure

I do not know whether this is the right place for posting this problem. But for several months I have no solution to this problem.
Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that ...

**2**

votes

**0**answers

31 views

### Solving the inverse of a matrix under a uniform distribution

I am looking to solve the following equation:
$$\left(\begin{array}{cc}{ g_{11}} & { g_{12}} \\ { g_{21}} & { g_{22}}\end{array}\right)=\int_{a}^{b} \frac{1}{b-a}\left(\begin{array}{cc}{- g_{...

**0**

votes

**1**answer

171 views

### Logarithm of an integral involving generalized real binomial coefficients

I could not find a closed form for this integral although I think it should have been studied.
What is a good approximation to $I$ in
$$I=\ln\Bigg(\int_{0}^y\binom{2m}{m(1+x)}dx\Bigg)$$ where $0\...

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votes

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57 views

### Derivative of a convolution integral of the following type?

I'm looking to find the derivative of a convolution integral of the following form:
\begin{equation}
\frac{d}{dr}((G(r,t)*f(t)) = \frac{d}{dr} (\int_{-\infty}^{\infty} G(r,t-\tau)f(\tau) d\tau)
\end{...

**1**

vote

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204 views

### Integral of matrix determinant with respect to Lebesgue measure

$\newcommand\norm[1]{\lVert#1\rVert}
\newcommand\opnorm[1]{\norm{#1}_{\text{op}}}
\newcommand\Frnorm[1]{\norm{#1}_{\text F}}$Define
\begin{align*}
S_t=\{
(A,B)\in\mathbb{R}^{n\times n}\times\...

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vote

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142 views

### Prove the following property about natural integral

Natural integral is the distinguished antiderivative of a function that can be understood as an analytic continuation of consecutive derivatives of a function towards $-1$th order. It is defined as
$...

**3**

votes

**2**answers

143 views

### A definite integral related to sample variances of bivariate Gaussians

This integral is needed to obtain the joint
distribution of the sample variances of a random sample from a bivariate
Gaussian distribution. For details on the joint distribution of the sample
means, ...

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votes

**0**answers

89 views

### Comparing an integral to zero, by slicing and stacking

Let $f \colon [0,1] \rightarrow {\mathbb R}$ be a nice function -- real-analytic, or maybe definable in some o-minimal structure, let's say. Let $0 < \alpha_1 < \cdots < \alpha_n < 1$ be ...

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votes

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359 views

### A Curved/Warped Version of Fubini's Theorem

I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.
Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ...

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vote

**0**answers

98 views

### |Evaluating integral on $ \mathbb S^{d-1}$

I am trying to evaluate the following integral:
$$ \int_{\mathbb S^{d-1}} \exp \bigg(-\frac{(1+x\cdot y)^2}{\|x+y\|^2} \bigg) \ dx $$
for $x,y \in \mathbb R^d$. Does anyone know a solution or an ...

**1**

vote

**1**answer

185 views

### Integration on sphere $\mathbb{S}^{d-1}$ for $d$ large — Change of variables

I'm trying to integrate a function over two vectors which lie on the surface of the unit sphere in D dimensions. The function depends only on the difference between the two vectors, and their dot ...

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votes

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311 views

### Meaning of divergent integrals

In quantum field theory, one usually encounters divergent integral when one calculates loop functions, such as the integral $\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}$ which is divergent. ...

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votes

**1**answer

195 views

### An integral of $|\sin(x)\cos(nx)^{-2/n}|$ from $-\pi$ to $\pi$

For an integer $n \geq 3$, define
$$A_n = \int\limits_{-\pi}^\pi\frac{|\sin(x)|}{|\cos(nx)|^{2/n}}dx.$$
It is a fact that $A_n$ is finite for all such $n$. I am interested in the behaviour of a ...

**1**

vote

**1**answer

404 views

### What are the consequences if we could express tangent via logarithm in an algebraic system? [closed]

Working on an algebra of divergent integrals I came to the following relation:
If $\tau=\int_0^\infty dx$ then
$$\ln (\tau+a)=\int_{0}^\infty \psi'(x+1/2+a)dx$$
and this directly gives the following ...

**1**

vote

**0**answers

140 views

### Solving an equation of function

How to solve, or at least how to proceed to solve, the following equation for $g(u)$
$$\int_0^{\infty} \{1-\cos(2\pi uh)\} g(u)du = (1+h^{\alpha})^{\beta/\alpha} -1?$$
Here $0<\alpha\leq2$ and $-\...

**6**

votes

**1**answer

480 views

### Possible application of divergence Theorem?

suppose that $f \in C^1 (\mathbb{R}^{N+1},\mathbb{R})$. It's well known that if all his points are regular points i.e.
$$\nabla f (x) \neq 0 \; \; \; \forall x \in \mathbb{R}^{N+1}$$
then, for every ...

**49**

votes

**4**answers

5k views

### A historical mystery : Poincaré’s silence on Lebesgue integral and measure theory?

Lebesgue published his celebrated integral in 1901-1902. Poincaré passed away in 1912, at full mathematical power.
Of course, Lebesgue and Poincaré knew each other, they even met on several occasions ...

**0**

votes

**0**answers

67 views

### Does $L^1$ convergence preserve the regularity of this sequence of functions?

Let $f_n$ be a sequence of $L^1(]0,1[)$ functions such that $f_n$ is non-decreasing, at least left-continuous, $f_n(0^+) <0$, $f_n(1^-) >0$, for all $n \in \mathbb N$. This sequence converges
$...

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votes

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145 views

### Is it true that the quantile function of an $L^1$ random variable is $L^2(]0,1[)$?

Let $(\Omega, \mathcal A, P)$ be a probability space. Let $X:\Omega \rightarrow \mathbb R$ be an $L^1(\Omega, \mathcal A, P)$ random variable.
We define the distribution function of $X$ by
$$F(x) = ...

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vote

**2**answers

67 views

### Opial type inequalities

Let $x(t)\in C^1[0,h]$ be such that $x(0)=x(h)=0$ and $x(t)$ in (0,h) ,then the following inequality
$ \int^h_0 |x(t)x^{'}(t)|dt \leq \frac{h}{4}\int^h_0(x^{'}(t))^2dt$
my question: I would like ...

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votes

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107 views

### Bayesian Bandits - What's the probability that choice K is the best?

I have $K$ very unfair coins. I don't know how unfair they are, but they all seem to have different probabilities of landing heads. I'd like to figure out which one is best as quickly as possible.
...

**0**

votes

**1**answer

171 views

### All roots of a polynomial are inside the unit circle [closed]

How to compute the following integral?
$\int_0^1 P_j(x)dx$, where $P_j(x) = \prod\limits_{i=0,i\ne j}^n(x+i)$?
I met this problem when doing estimation on the roots of the following polynomial.
All ...

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vote

**0**answers

55 views

### Daniell integral of “generalized (of some sort)” functions?

Let $E$ be a (Dedekind $\sigma$-complete) Riesz space and $H\subseteq E$ a subspace. A Daniell integral $I\colon H\to\mathbb R$ is defined to be a positive linear functional which is continuous with ...

**5**

votes

**1**answer

278 views

### Can I cover a compact set by balls {B} such that {2B} has bounded overlap?

Suppose I have a compact set $K \subset B_1(0) \subset \mathbb{R}^n$. Can I always find a family of open balls $\{B_{r_j}(x_j)\}$ such that
$x_j \in K$ and $B_{r_j}(x_j) \subset B_1(0)$ for each $j$;
...

**1**

vote

**1**answer

228 views

### If $h$ is a decreasing function then $\psi$ is an increasing function

Let $h : [0,1] \to [0,1]$ be a $\mathcal{C}^1$ function such that $h'(x)<0$ for all $x \in (0,1)$. Consider the function
$$
\psi(x) = \frac{\int_0^1yh(|x-y|) dy}{\int_0^1 h(|x-y|)dy}
$$
I am trying ...

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vote

**0**answers

132 views

### An integral involving many exponential terms with quadratic exponents in the denominator

Given $k$ points $\{p_1,\cdots, p_k\}$ in $\mathbb{R}^n$ and positive constants $r_1, ..., r_k$ and another positive constant $\alpha>0$. Is there a way to compute/approximate the following (...

**1**

vote

**1**answer

256 views

### Is this relation between divergent intergals justifiable?

Graf's book on hyperfunction theory says (page $36$) that
$$\frac1{(x-i0)^n}=\frac{(-1)^{n-1}\pi i}{(n-1)!}\delta^{(n-1)}(x)+\operatorname{fp}\frac1{x^n},$$
while the table of Fourier transforms ...

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votes

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164 views

### Dominated convergence Theorem

I am struggling to understand the proof in the paper, Learning Temporal Evolution of Spatial Dependence with
Generalized Spatiotemporal Gaussian Process Models.
Theorem 2.1 in the page 33 uses ...

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votes

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178 views

### Is there a practical application of natural integral or differintegral?

The following formulas give natural differintegral (that is one with naturally fixed integration constant):
$$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
$$...

**1**

vote

**1**answer

58 views

### Perform certain constrained integrations over an ordered subsection of a 3-simplex, yielding “absolute separability” probabilities

Let us order the four points $\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq 0$ of a 3-simplex, $\lambda_1+\lambda_2+\lambda_3+\lambda_4=1$, giving us a subsection $L$.
Integration over $...

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votes

**1**answer

144 views

### Taylor expension of a simple integral [closed]

I'm trying to derive some weights expression for a boosting algorithm on a L2-ISE loss function, and i have trouble with the taylor expension.
Suppose that $f$ and $g$ are two densities from $\...

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90 views

### Change variable in integration with symmetry

Not sure if I can ask such fundamental problem here.
Let
$G$ be a finite group, $\sigma \in G$. Consider linear group actions fo $G$ on $\mathbb{R}^n$.
$\sigma(K)=\{x\in \mathbb{R}^n: \sigma^{-1}(...

**2**

votes

**1**answer

106 views

### Example of evaluation of $\int_0^1\left(\sum_{k=0}^n (f(x))^k\right)^{\alpha}dx$, for some choice of $f(x)$ satisfying certain requirements

Let $0<\alpha\leq\frac{1}{2}$ a fixed real number. I wondered if it is possible to evaluate the sequence of definite integrals $$\int_0^1\left(\sum_{k=0}^n (f(x))^k\right)^{\alpha}dx\tag{1}$$
for ...

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votes

**3**answers

296 views

### Density of a functional space

Let $D$ be a bounded domain of $\mathbb{R}^n$ with smooth boundary $\partial D$. Is the following subspace dense in space $L^2(D)\times L^2(\partial D)$:
$$\{(f,f\rvert_{\partial D}) : f\in C^\infty(\...

**3**

votes

**1**answer

419 views

### Risch's algorithm for symbolic integration and its variations

I want to explore symbolic integration, but for this I initially need to imagine what are the algorithmic achievements in this area today, so I have some questions about Risch's algorithm and all its ...

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votes

**1**answer

102 views

### Sufficient condition for function of conditional probability density to be increasing

Let $Y$ and $W$ be two jointly distributed random variables; $Y$ takes values on $(y_1,y_2)$ and $W$ takes values on $(w_1,w_2)$. The conditional probability density of $W$ given $Y$ is given by $f_{W|...

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votes

**2**answers

222 views

### Express $\int_0^{\pi/2}\{ \operatorname{gd}^{-1}(x)\}dx$ as series of special functions, with $\operatorname{gd}^{-1}(z)$ the inverse Gudermannian

I know that in the literature there are a lot of integrals involving the fractional part (and other floor and ceiling functions). For some of these integrals is provide its evaluation as a series or ...

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vote

**1**answer

68 views

### Inequalities for the Gudermannian function of the type $\operatorname{gd}(x)\operatorname{gd}\left(\frac{1}{x}\right)<\text{upper bound}$, where $x>0$

After I've read the solution of Problem 4327 (see [1]) I wondered if an inequality for a similar upper bound in the RHS is feasible for the known as Gudermannian function
$$\operatorname{gd}(x)\...

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votes

**1**answer

161 views

### Uniform sampling on a Riemannian manifold via tangent space and exponential map

Given a Riemannian manifold $(\mathcal{M}, \{g_x\}_{x \in \mathcal{M}})$ and a fixed point $x \in \mathcal{M}$, does the following procedure yield uniform samples from $\{y \in \mathcal{M} : d_\...

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vote

**1**answer

221 views

### Examples of Steffensen's inequality at undergraduated level studies

I've known few days ago the known as Steffensen's inequality, see the article Steffensen's inequality from Wolfram MathWorld and the cited bibliography. It seems that there are applications (I don't ...

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votes

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919 views

### Challenge: Non-Gaussian quartic integral and path integral in Quantum field theory

(1) It is well-known that we can get a Gaussian integral of this type, where $x$ is in $\mathbb{R}$:
$$
\int_{-\infty}^{\infty} dx e^{-ax^2}=\sqrt{(2\pi)/a}. \tag{i}
$$
We can generalize this ...

**1**

vote

**1**answer

145 views

### A marginal space splitting $\{ \psi \}^{\perp}$

Let $\psi \in L^2(\mathbb R^2,\mathbb C)$. Is there a continuous projection from $\{ \psi \}^{\perp}$ onto
$$
\left\{ \varphi \in L^2(\mathbb R^2) \:\:\Big| \int \overline{\psi}(x,y) \varphi(x,y)\...

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vote

**1**answer

87 views

### On a type of sequence of integrals inspired in the Borwein integral and an integral due to Furdui

This post is inspired in the Borwein integral, and in a problem proposed by Ovidiu Furdui in Crux Mathematicorum, that is the Problem 3707, in page 151.
I've considered integrals of the form $$\int_0^...

**5**

votes

**1**answer

2k views

### A rather curious equality: is this true?

I came across (coincidentally) two integral evaluations, which seem to agree according to numerical tests. It did not seem easy to convert one into the other.
QUESTION. Is this true?
$$\int_0^1\...

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vote

**0**answers

90 views

### Reparametrization of a closed curve that balances sum of first derivatives

(Question in the yellow box below.)
A few weeks ago I was wondering about the existence of a scalar function $f(s): S^1 \rightarrow \mathbb{R}$ and a turning angle $\phi(s):S^1 \rightarrow \mathbb{R}/...

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vote

**1**answer

109 views

### Limiting value of definite integral [closed]

$$I = \int_{-4}^4 e^{in\pi x/4}\frac{\sinh(b\pi/4)}{\sin^2\left(\frac{a-x}{8/\pi}\right)+\sinh^2(b\pi/8)}\,dx$$
I am unable to integrate the above equation when when $b$ tends to $0$, because of a ...

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vote

**2**answers

357 views

### Prove that a certain integration yields the value $\frac{7}{9}$

Numerical methods surely indicate that $\int_0^{\frac{1}{3}} 2 \sqrt{9 x+1} \sqrt{21 x-4 \sqrt{3} \sqrt{x (9 x+1)}+1} \left(4 \sqrt{3} \sqrt{x (9
x+1)}+1\right) \, dx= \frac{7}{9}$.
Can this be ...

**5**

votes

**1**answer

177 views

### Triangle inequality for Ito integral?

For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say
$$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$
Now if ...

**0**

votes

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141 views

### Weak derivative under the integral sign

Let $\Omega$ be a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$ and $u:[0,\infty)\times \Omega\to \mathbb{R}$ be a smooth function (for example a smooth solution to a PDE). Thus the ...