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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152 views

Evaluating $\int \limits_{a}^{\infty} \frac{\exp\left(-ax\right)}{\log(x)\left(c+x\right)^2}dx$

I have the following integral $$\int \limits_{a}^{\infty} \frac{\exp\left(-ax\right)}{\log(x)\left(c+x\right)^2} dx$$ that I do not know how to evaluate. Could you please give me a hint? Thanks in ...
3
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102 views

integral equation solution for two functions $ f(x) $ and $ g(x) $ and see if they are related

given two functios $ f(x) $ and $ g(x) $ related by $$\frac{ \Gamma(s-1/2)}{\Gamma(s) \sqrt{ \pi}}\int_{0}^{\infty}dx \frac{g(x)dx}{(x+y)^{s-1/2}}=\int_{0}^{\infty}dx \frac{f(x)dx}{(x+y)^{s}}$$ what ...
3
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193 views

Integral of a product

Given the following integral: $$I(N)=\int_{0}^{t}\prod_{k=1}^{N}\sin(k\omega \tau)d\tau$$ does someone know if is it possible to find the solution of $I(N)$ in a closed form? I'm able to find the ...
3
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59 views

How to approach an integral over $g(\cos(t))$ from $0$ to $2\pi$, where $g(x)$ is nasty?

For notational convenience, let $f(t) = a^2 + 2 a b \cdot \cos(t) + b^2$, where $a,b$ are both positive real constants and $t$ will be the integrand of the integral, which is supposed to be carried ...
3
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174 views

Pontryagin's maximum principle problem, minimising an integral

I am really stuck on the following: Use Pontryagin's maximum principle to show that $\pi$ is the minimum value of the integral $\frac{1}{2} \int _{0} ^{1} u^{2} + v^{2} \,dt$ subject to constraints ...
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252 views

Taking derivative below an integral

I am trying to solve the following question: If $t>0$, then \begin{align*} \int_{0}^{+\infty} e^{-tx} \; dx = \frac{1}{t} \end{align*} Moreover, if $t \geq a > 0$, then $e^{-tx} \leq ...
3
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135 views

What is the Dunford Integral and why is it useful?

Wikipedia defines the Pettis Integral for Banach space valued functions on a measure space by duality. Apparently there is a Dunford integral which specializes to the Pettis integral. What is its ...
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332 views

Lebesgue Line Integrals - Parametric Change of Variables

Consider the following Lebesgue integral in $\mathbb{R}^n$ $$ \int_C f(x) dx $$ Where $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is measurable and $C$ is a measurable subset of $\mathbb{R}^n$ that ...
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84 views

Asymptotics of Riemann-Lebesgue type integral

How to show that for $u \in L_{\mathbb{C}}^2$ and $a>0$, $$\int_0^a u(t) \sin{\sqrt{\lambda}t} \,dt = o(e^{|Im\sqrt{\lambda}|a}),\text{ as } |\lambda| \rightarrow \infty$$ Note that $\lambda$ ...
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49 views

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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226 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: ...
3
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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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174 views

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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224 views

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: $ ...
2
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45 views

Interesting examples of switching limit and integral

We learn many theorems regarding the relationship of limit and integral (Dominated/ Monotone Convergence, Fatou, Semicontinuity of norms, etc...). As I'm working on my research, I find that I often ...
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22 views

what are and why are sine and cosine modulated integrals used?

I have found the definition of the following formulas in a paper, where they are called sine and cosine modulated integrals. $y$ is a signal with a strong periodic component of frequency $N\Omega$ ...
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46 views

Proving a set is of measure zero.

Let $C\subset A\times B$ be a set of content zero. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B: (x,y)\in C\}$ is not of content zero. Show that $A'$ is a set of measure zero. ...
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66 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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103 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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54 views

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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62 views

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 $|\omega|<\omega_b$}, ...
2
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58 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 $b$, ...
2
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46 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 ...
2
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62 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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40 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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85 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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53 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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35 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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0answers
39 views

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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26 views

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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32 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: $$ ...
2
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0answers
37 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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20 views

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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0answers
83 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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35 views

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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22 views

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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25 views

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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39 views

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?
2
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29 views

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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50 views

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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19 views

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 ...
2
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0answers
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 ...
2
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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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0answers
47 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 ...