# 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, ...

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### 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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### 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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### 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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### |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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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 ...
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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? $$\int_0^1\... 0answers 90 views ### Reparametrization of a closed curve that balances sum of first derivatives (Question in the yellow box below.) A few weeks ago I was wondering about the existence of a scalar function f(s): S^1 \rightarrow \mathbb{R} and a turning angle \phi(s):S^1 \rightarrow \mathbb{R}/... 1answer 109 views ### Limiting value of definite integral [closed]$$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$$I am unable to integrate the above equation when when b tends to 0, because of a ... 2answers 357 views ### Prove that a certain integration yields the value \frac{7}{9} Numerical methods surely indicate that \int_0^{\frac{1}{3}} 2 \sqrt{9 x+1} \sqrt{21 x-4 \sqrt{3} \sqrt{x (9 x+1)}+1} \left(4 \sqrt{3} \sqrt{x (9 x+1)}+1\right) \, dx= \frac{7}{9}. Can this be ... 1answer 177 views ### Triangle inequality for Ito integral? For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say$$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds Now if ...
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 ...