All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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understanding of the rank-2 tensor as an integral kernel

Let $G\subset{\mathbb R}^m$ be a nonempty compact and Jordan measurable set that coincides with the closure of its interior. Let $K:G\times G\to{\mathbb C} $ be a continuous function. Then we can ...
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224 views

Are Specific Facts about the Riemann Integral Logically Required?

This question is somewhat in the spirit of this one in that I am trying to understand the most efficient path to the major integral theorems (Fubini, change of variables, etc). My question is this: ...
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100 views

Integral when variable of integration is a multivariable function

I recently ran into a kind of integral that I've never encountered before. How should the following integral be expressed as a "normal" double integral? $\iint \mathrm d f(u,v)$ where ...
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56 views

Integrability of $\frac{k-1}{2(1+|x|^k)}$

Is the following function known to be integrable? It is supposed to be a probability density function, i.e., integrates to one. However, it leaves the online Mathematica integrator stumped: ...
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Change of variables in line integral with abs. value

Let $\gamma : I \rightarrow \mathbb C$ be a path. Let $g: \mathbb C \rightarrow \mathbb C$ be a biholomorphic map. Let $f$ be a holomorphic function. Consider the integral $$ \int_{g\circ \gamma} ...
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Series of nested double integrals

This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals $$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
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299 views

What is the expected value of modified Dirichlet distribution? (integration problem)

It is easy to produce a random variable with Dirichlet distribution using Gamma variables with the same scale parameter. If: $ X_i \sim \text{Gamma}(\alpha_i, \beta) $ Then: $ ...
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58 views

How to evaluate the integral $\int e^{ipx}e^{ipx} d^{3}x = 0$

I am embarrassed to ask this question. But I came across the following in a physics book: $$\int e^{ipx}e^{ipx} d^{3}x = 0$$ $d^{3}x = dydydz$, as @Semiclassical shows below. This came up in the ...
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98 views

Definite trigonometric integral

This question is motivated by Iterative Mean, Covariance Algorithm Convergence: Is there a closed form for the integral $$ \int_0^{2 \pi} ...
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36 views

How do textbooks display the mean by integration

My question may be expressed with an example I'm most familiar with.. In many ocean science text books, the mean density of the water column is expressed as $${\widehat\rho}=\int^0_{-h}\rho(z)dz$$ ...
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Contour integral: different answers with different contours

Good day to everyone. I have a following contour integral problem. I have to find a solution for the integral $$\underset{\gamma_r }{\oint }\frac{e^{\lambda s} }{(1-s) s^{a-b} \left(s-\theta ...
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Simplifying a Vector Integral

While reading the book - Cercignani, Theory and Applications of Boltzmann Transport Equation (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , \xi_l$ ...
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53 views

Fourier transform of a sinusoidal function

Let us consider following table which I want to calculate myself $$ x(t)=\frac{\sin(\omega_bt)}{\pi t}\quad\iff\quad X(j\omega)= \begin{cases} 1 & \text{if $|\omega|<\omega_b$}, ...
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56 views

Evaluate $\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx$

Is there a closed form for the integral $$\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx?$$ where $\lambda>0$, $a>0$, $d>0$ and where $b$, ...
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45 views

Risch Algorithm for trigonometric functions

I've implemented the transcendental Risch integration algorithm for logarithmic and exponential extensions ('classical Risch'). If I want to integrate functions containing trig-functions I have to ...
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53 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
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72 views

An integral problem related to matrix determinant

I am stuck in an integral problem: ...
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39 views

Notation of an infinite integral

I'm confused regarding the notation of the integral I've underlined in green. Does this mean that the integral over any range is $\infty$?
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84 views

Calculate the Gauss integral without first squaring it

We know that the integral $$I = \int_{-\infty}^{\infty} \mathrm{d}x e^{-x^2}$$ can be calculated by first squaring it and then treat it as a 2-d integral in the plane and integrate it in polar ...
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51 views

inverse laplace transform of $$p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$$

I used the Residue theorem to solve this problem. But, I could not obtain the solution given by $$\mathscr{L}^{-1}\left( { p^{-3/2}e^{-\sqrt{pa}}\over{2\sqrt{2}}} [\cos(\sqrt{ap})+\sin(\sqrt{ap})] ...
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34 views

Difficult integrals, do they converge, show there's no dependence on parameters.

I am trying to figure out whether these integrals: a) $$\int_{\mathbb R^2}{{\rm d}\xi \over \left\vert\vphantom{\Large A}\,\log\left(\left\vert\,x - \xi\,\right\vert\right) -\log\left(\left\vert\,y ...
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Are there high performance computing applications for symbolic integration?

Currently there are a number of applications for numerical integration in applied mathematics and physics. Many of these are integral transforms (often Fourier or Laplace), or solving definite ...
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47 views

Complex Contour Integration - Complex Analysis

I'm just practising for my upcoming exam, and I've come across a question I'm having a bit of difficulty with. I've been asked to show the following; $$\int_{0}^{\infty} \frac{dz}{\cosh(z)} = ...
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Prove Green's theorem for circles

So the problem is in the title. The rules are that I can't split the circle into "rectangles" and I can't use pull-back. I tried to do something similar to the proof on unit squares. The problem is ...
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114 views

What is ${\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$?

This is a new integral that I propose to evaluate in closed form: $$ {\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$$ where $\log (z)$ denotes the principal value of ...
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29 views

Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function: $$ \mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right) $$ $z \in \Bbb{C}$ which also has the following integral representation: $$ ...
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36 views

Can someone give an example of a function that is not Henstock-Kurzweil/gauge integrable?

I am looking for an example of a function that is not Henstock-Kurzweil integrable. Can anybody help me?
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help finding this surface integral

Given that $$S : z = xe^y, \hspace{1mm} 0 \leq x\leq 1, \hspace{1mm} 0 \leq y\leq 1 $$ Find $$\int\int_S \left(x^2+y^2+z^2\right)dS$$ upto four decimal places
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48 views

Line integral - Stokes's theorem

Let $\gamma_1$ be the intersection curve between the surface $x^2+(y+z)+z^2=1$ and the plane $z=x$ and let $\gamma_2$ be $\gamma_1$ with $x\geq0$,$y\geq0$. Calculate ...
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80 views

Improper integral $ \int_{0}^{\pi/2} \frac {\sqrt{x} } { (\pi/2-x)(1-\exp(\tan(x))) } dx $

I need to calculate following improper integral: $$ \int_{0}^{\pi/2} \frac {\sqrt{x} } { (\pi/2-x)(1-e^{\tan(x)}) } dx $$ I have no idea where to start.
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Finding flux of a vector field $\mathbf{F}$ across a surface bounded by an unknown function.

A solid $\Gamma$ in $\mathbb{R}^{3}$ is bounded by $$0 \leq x \leq 1, \hspace{.5cm} 0 \leq y \leq 1, \hspace{.5cm} 0 \leq z \leq g(x, y),$$ where $z = g(x, y)$ is an unknown differentiable surface. ...
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A proof regarding Fourier-Polynoms

I want to prove the following: Let $f:\mathbb{R}\rightarrow \mathbb{C}$ so that $f \big |_{[0,2\pi]}$ is integrable. Let $V$ be the vectorspace of all $2\pi$-periodic functions and $U \subset V$ be ...
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Greens function of a uniformly charged sphere

The potential $\phi(\boldsymbol{x})$ satisfies $\nabla^2\phi=f$ It may be shown that by defining an appropriate Green's function $g(\boldsymbol{x},\boldsymbol{\xi})$ that ...
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Is there a way besides integration by parts to solve this integral?

$$\int_{0}^{2\pi} -10\cos^9(t)\sin^4(t)t^4\,dt$$ Maybe a formula for this form or something?
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Is little-o preserved under integration and derivation of another variable?

Given an integrable function $g:\mathbb{R}\longrightarrow\mathbb{R}$, and a function $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ such that $f(x,y)=o(x^{-1})$ when $x\rightarrow\infty$, i.e. ...
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Multivariable Integral Calculus help

I have two questions. First: Is my proof "strong" enough? I am being asked to prove that $$\int_{0}^\infty\int_{0}^x e^{-sx}f(x-y,y) dydx = \int_{0}^\infty\int_{0}^\infty e^{-s(u+v)}f(u,v) dudv$$ ...
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integrate the square of angular distance from the node of a spherical triangle

Guessab, Noouisser, and Schmeisser "A Definiteness Theory for Cubature Formulae of Order Two", Constructive Approximation (2006)24:263-288 Define a quantity $R[||\cdot||^2]$ which is $$\sum_{i=1}^N ...
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29 views

Monte Carlo with error on individual samples

I'm performing a Monte Carlo integration where the individual samples have an error, and I'm wondering how to estimate the final error. Some more detail: The integral E I'm after is estimated in the ...
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28 views

An abstract integration problem from a mathematical finance calibration problem

I would massively appreciate help on this problem which relates to me trying to calibrate my financial model to market data. It can be stated without reference to any finance, this is my abstract ...
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46 views

Integral of a derivative

Let $f\colon[a,b]\to\mathbb R$ be differentiable. Analysis provides us with several sufficient conditions for the formula $$ \int_a^bf'=f(b)-f(a) $$ to be true, like continuity (or just Riemann ...
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49 views

Calculating this integral: $I=\int_{0}^{\infty}(\log t)\,(\tan^2t)\,\mathrm{d}t$

How to calculate this integral? $$I=\int_{X_0}^{X}(\log t)\,(\tan^2t)\,\mathrm{d}t.$$ I tried integrate by parts and I found something related to: $$J=\int_{X_0}^{X}\dfrac{\log\cos ...
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Would like to calculate the following limit:

$\lim_{n \to \infty} \int_0^1 \arcsin (\sin(nx)) dx$ I think the answer is 0, but can't prove it.
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Was Euler really the first who proposed the streamline integration principle?

I am interested who first came up with line integration in a vector field to construct lines. So far I think (based on Wikipedia) it was Euler with his book: Institutionum calculi integralis. Do you ...
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98 views

An Integral possibly related to Legendre polynomials

Consider the integral $$\int_0^1\frac{(t^2-1)^a}{(t-u)^{b+1}}dz$$ where $b\gg a$, with $a,b$ integers and $u>1$. I know you can write this integral as the sum of two hypergeometric functions but ...
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About differentiation under the integral sign

I would like to ask something related to the application of the differentiation under the integral sign (Leibniz rule) given by ...
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Question about convergence of improper integral

Could you give me some hint how to solve this problem: Suppose $f$ is continuous on $(0,1]$ and there is $M$ such as $\left|\int_x^1f(t)\, dt \right|\le M$. Prove that $\int_0^1f(x)\, dx$ converges ...
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38 views

Complex integral and parametrization of a circle

I am trying to compute the following integral of $$\int \frac{1}{z^3+3} dz$$ over a circle of radius $2$, centerd at $(2,0)$. Thus I am trying to compute the residue and have found that the function ...
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47 views

Is there a generalization of integration by parts?

here is what i concerned: there are $u(x)$ and $v(x)$ in the original integration by part formula, what if the integral involve with one more function $w(x)$. Second of all, i know that there are ...
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Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
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Improper integrals in curve length

I am supposed to find the length of curve of the following: $ y = \sqrt{2-x^2}$ ; $0\le x\le 1$ $y =\ln(\cos x) $; $0\le x\le \frac{\pi}{3}$ I followed the directions found from this question : ...