Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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Origin of the Integral (Theory Behind It - How it came about)?

How exactly was the integral derived? Like similarly to how the difference quotient explains where the derivative came from, what can we use to explain the origins of the integral? Like how does ...
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76 views

When can we use Differentiation under the Integral sign?

Let me elaborate, 'Feynman's' trick ranks up in the top ten on most people's list, right behind contour integration, for best ways to evaluate definite integrals. However, unlike contour integration ...
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61 views

how to solve this inverse fourier $ f(x) =\int^{\infty}_{-\infty} 1/\sqrt{2\pi}\ e^{-2\pi^2/s^2} e^{ i \ s\ x}ds$

I have two functions f(x) and f(s). f(s) is the fourier transform of f(x) and tends to $$e^{-2\pi^2/s^2}$$ I need to take inverse transform of this f(s) to get to f(x). (i need to prove f(x) tends to ...
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163 views

Using a sequence of measures to create simple functions which approximate the Radon-Nikodym derivative of the limiting measure

I have a bunch of discrete probability measures with finite support: $\mu_1,\mu_2,\dots$, which strongly converge to an absolutely continuous probability measure $\mu$ in $\mathbf{R}^2$. That is, for ...
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51 views

Do we have a inverse Laplace transform of $\frac{1}{\arctan s}$

Do we have a closed form of this seemingly very simple inverse transformation? If no closed form, what about its asymptotic form? Does this satisfies the criterion to have its inverse ...
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58 views

Simpson's Rule for Double Integrals

Simpson's Rule for double integrals: $$\int_a^b\int_c^df(x,y) dx dy$$ is given by $$S_{mn}=\frac{(b-a)(d-c)}{9mn} \sum_{i,j=0,0}^{m,n} W_{i+1,j+1} f(x_i,y_j) $$ where: $$W= \begin{pmatrix} ...
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30 views

First mean value theorem for integration and Lebesgue measureability

According to first mean value theorem for integration, if $G \ : \ [a,b] \to \mathbb{R}$ is a continuous function, there exists $x \in (a,b)$ such that $$\int_a^b G(t) dt = G(x)(b-a)$$ Assume $G$ is ...
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41 views

How to Solve this Improper Integral with six poles?

I'm trying to solve the following integral, where $a>0$, $b>0$, $y\in\mathbb{R}$ and $z\in\mathbb{R}$ are given constants: $$ \int_{-\infty}^{0} \left[ ...
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49 views

Double integral of symmetric polylogarithmic function over rectangular region

This question was inspired by M.N.C.E.'s wonderful response here. While exploring the possibility of generalizing his result, I found that a significant part of the problem reduced to evaluating the ...
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62 views

Solution of $\int_0^{\pi} \frac{ y \cos y}{s^2+y^2} dy$

Is there a solution for the following integral (even in terms of Bessel or Struve functions)? $$ \int_0^{\pi} \frac{ y \cos(y)}{s^2+y^2} \,dy $$
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56 views

Integral involving Whittaker function

Consider the following integral: $$ \int_1^{\infty} \frac{e^{u/2}}{u}[-\mathrm{Ei}(-u)]\,W_{1,\imath p}(u)\,du, $$ where $\imath=\sqrt{-1}$ and $p>0$ selected so that $W_{1,\imath p}(1)=0$; here ...
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55 views

Strange triple integral of an inverse function

Let $$ \Omega(a, b, c) = \min\left\{\theta\ge0\ \text{s.t.}\ \tan(a\theta) + \tan(b\theta) + \tan(c\theta) = 1\right\} $$ What is the value of the following integral $$ I = ...
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77 views

Fast Hankel Transform

Can someone please explain what would be the expression for weights(Ho) in a Fast Hankel Transform.I found this in a paper and could not find any satisfactory answers .
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905 views

Solving an integral for a characteristic function

For $L>0, H>0,\alpha>0,\sigma>0,$ $$f(t)=\int_L^H \frac{ e^{i t x} \alpha H \left(\frac{\sigma -H \log \left(\frac{H-x}{H-L}\right)}{\sigma }\right)^{-\alpha -1}}{\sigma (H-x)} \, ...
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96 views

How to evaluate $\int_1^\infty \frac{1}{z} e^{-\left(\frac{z-1}{b}\right)^{\frac{1}{a}}} dz$?

I am really stuck on this integral: \begin{equation} \int_1^\infty \frac{1}{z} e^{-\left(\frac{z-1}{b}\right)^{\frac{1}{a}}} dz \end{equation} where $a,b$ are real constants. Is it a special ...
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23 views

Stieltjes Integral - If $f, f^2, g, g^2\in R(\alpha)$ for an arbitrary integrator $\alpha$, then is $fg\in R(\alpha)$

My question is if $f, f^2, g, g^2\in R(\alpha)$ on $[a,b]$ for an arbitrary integrator $\alpha$, then is $fg\in R(\alpha)$ as well? This question stemmed from a problem in Apostol's Analysis, in ...
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58 views

Is there a spherical coordinates system for vectors of complex numbers?

Suppose I have a scalar field $f(\vec{x})$, where $\vec{x}\in\mathbb{R}_3$, and I wish to average $f$ over a sphere $|\vec{x}|=R$: $\displaystyle\langle f\rangle_{R} = \frac{\int_{S} f(\vec{x})\, ...
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50 views

Implementing the Risch algorithm to integrate $\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}$

Following the work of Andreas Wurfl i am trying to implement the Risch algorithm on $\int{\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}dx}$ following his method for extensions that are purely logarithmic, we ...
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76 views

If $y'=\frac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) $

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) = ?$ By integrating I am getting $$y = \ln (x+1)+C$$ I am stuck somewhat as it looks tricky from here. Any help ? Thanks!
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53 views

Change of variables in $\iiiint$ or $\idotsint$?

I apologize if this is a question that has been asked before, but I have seen that it is possible to change variables in single, double and triple integrals. Now what about quadruple integrals, or ...
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121 views

Visualising surface integrals

For a current problem I am working on, I have run into angular surface integrals, i.e. the differential solid angle $\text{d}\Omega$. Specifically the surface integrals are defined by ...
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55 views

closed form of improper integral

I would like to evaluate an integrale which depend on 2 parameters. The goal is to obtain an expression of the integrale depending on theses 2 parameters ($x_d$ and $z_d$) such as $f(x_d,z_d)= ...
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100 views

Graphs of interesting integrals of the form: $\int \sin^a(x^a)\cos^a(x^a)$

Here are a few graphs of the form:- $$\int \sin^a(x^a)\cos^a(x^a)dx$$ Where $a$ is an even, positive integer. $a = 2$ $a = 4$ $a = 6$ Now, a few graphs of the form:- $$\int ...
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43 views

Tauberian limit.

Let $0 < c_1 < \varphi(x) < c_2$ for positive $x$. Define $I_\alpha(T) = \int_0^T \varphi(t)t^\alpha dt$. If $\lim_{T \to \infty} I_{-1}(T)/\log(T) = C$, does it imply that $\lim_{T \to ...
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29 views

Riemannian Volume vs. Euclidean Volume

If $M$ is a Riemannian manifold and $\phi$ is a Borel measurable function into $M$, then what is the relationship between $\int_M \phi \,d\mu$ and $\int_{\mathbb{R}^d} \psi\circ \phi\, d\lambda$; ...
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30 views

How to recover complex function on $\mathbb C$ from integral equation?

Let $f:\mathbb C \to \mathbb C$ be a continuous function with the form $f(z)= z\tilde{f}(|z|)$ for all $z\in \mathbb C,$ where $\tilde{f}$ is a real function defined on $(0, \infty).$ We define ...
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63 views

Help me to figure out this integral

$$\int_{0}^{\infty} \frac{x^3\sin\left(\frac{1}{2}\pi x\right)}{e^{2\pi\sqrt{x}} - 1}~dx$$ I've been thinking a long time ,but I have no idea how to do it.
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69 views

Understanding why $\int_{-a}^a \sin^{100}x\,\mathrm dx = \frac{99}{100} \int_{-a}^a \sin^{98}x\,\mathrm dx$

In an answer here on Math.SE it is claimed that $$\int_{-a}^a \sin^{100}x\,\mathrm dx = \frac{99}{100} \int_{-a}^a \sin^{98}x\,\mathrm dx$$ but I don't understand how it could be. Since $$ ...
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45 views

A Step in the Proof of Green's Theorem

There is a step in the proof of Green's Theorem where we must combine the line integrals of each curve in the same direction. $$\oint\limits_C P(x,y)\,dx = \int_a^b P(x_1,y_1)\,dx + \int_b^a ...
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84 views

Equivalent Vitali Covering Properties for Differentiation Bases

N.B. In what follows below, we work exclusively with the Lebesgue measure on $\mathbb{R}^{n}$. Def 1 For each $x\in\mathbb{R}^{n}$, let $\mathcal{B}(x)$ be a collection of bounded open sets $R$ ...
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45 views

Center of Mass of objects with infinite length?

Suppose you have $f(x) = \frac{\sin(x)}{x}$ And you have that shape, find the center of mass of $f(x)$ in $x \in (-\infty, \infty)$ Is it possible considering $f(x)$ is an even function?
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42 views

How good is my approximation of this complicated sinc function? (plot included)

Part 1: The following function (for $N=256$) has the plot shown nexr $$ G(x) = \frac{1}{N}\text{exp} \bigg( j \frac{\pi}{2} \, x(N-1)\bigg) \frac{\sin (\frac{\pi N}{2} x)}{\sin (\frac{\pi}{2} x ...
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101 views

Any smart tricks to simplify my nasty integration?

I am trying to solve for the following unpleasant integral $$\int_{\gamma}^{\infty} \bigg[t- \int_{-2}^{2}\frac{ t \ f_X(x)}{1+N \ \big|G(x)\big|^2 \ t^{-3}}\ dx\bigg] \ dt$$ where $N$ is a ...
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48 views

Understanding averaging of symplectic matrices via Haar measure

In McDuff and Salamon's Intro. to Symplectic Topology (2nd edition), there's a proof that $U(n)$ is a maximal compact subgroup of $Sp(2n)$ which I'm trying to understand. The proof uses the Haar ...
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31 views

Issues proving a basis via wedge product

On a quiz I was given the problem" a series that is a basis for $[-1,1]$ is $ \sum_0^{\infty} c_n P_n $, where $ P_n $ is a polynomial and each polynomial $P_n$ is orthonormal to the others. Using the ...
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55 views

$\lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$

I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$? I thought to change this to an integral, namely ...
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76 views

how to calculate this line integral $\int_{0}^{2\pi} (16\sin^2 3t +16\cos^2 4t)\sqrt{(144\cos^2 3t +256\sin^2 4t)}dt$

I am working on a line integral to calculate the amount of chocolate to cover a pretzel. the density of the pretzel is given by this formula $\lambda=3(x^2+y^2)$ and the parameter equation of a ...
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49 views

Prove the function is integrable

For a point $x \in [1,2]$, define $f(x) = 0$ if $x$ is irrational and define $f(x)= \frac 1n$ if $x$ is rational and is expressed as $x = \frac mn$ for natural numbers $m$ & $n$ having no common ...
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66 views

Fatou's Lemma and Counting Measure

I have a vague problem in a Measure and Integration book here. They ask me to consider $\mu$ the counting measure in $\mathbb{N}$ and interpret Fatou's lemma, monotone and dominated convergence ...
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79 views

Klein-Gordon field commutator integral?

Consider a Klein-Gordon field $\phi$, which satisfies $$(\Box+ \omega_0^2)\phi=0$$ on points $x \equiv \{x_0,\vec{x}\},y\equiv \{y_0,\vec{y} \}$ of 4D Minkowski-spacetime. The field commutator is $$ ...
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63 views

Differentation uder the integral sign

Let $F(x)=\int_{\sin x}^{\cos x} e^{x\sqrt{1-y^2}} \, dy $. My task is to calculate $F'(x)$. My idea is to use http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign and I get: ...
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98 views

Integration involving square root function

First , i multiply numerator and denominator by (2-x-x^2)^(1/2), then I split the integral into 2 parts , using trig substitution , part 2 is easy to be integrated .. but when I tried to ...
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107 views

Is there a way to exploit local redundancy in a function to speed up Monte Carlo integration?

In every Monte Carlo method I've ever seen, $f$ must be recomputed from scratch for each point that is (somehow randomly) selected to contribute to the overall integral. However, most functions have ...
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68 views

Solving system of first-order PDEs with Frobenius theorem

I've been stuck trying to solve this system: $$\ \frac{\partial u}{\partial x} = \frac{-2xy^2}{u} + 3y $$ $$\ \frac{\partial u}{\partial y} = \frac{-2x^2y}{u} + 3x $$ Which must satisfy $ \ u(0,0) = ...
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27 views

Finding the Area of a Torus-like surface

I'm trying to find out the Area of the following surface: Let $C$ be the curve associated to a regular, simple path $\theta:[0,l]\rightarrow \Bbb R^2 $; also assume that ...
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80 views

Prove that primitives of $\frac{x^3}{{\rm e}^x - 1}$ have no closed form in terms of elementary functions

It is known the following indefinite integral $$\int \frac{x^3}{{\rm e}^x - 1} dx$$ cannot be evaluated in closed form in terms of any of the elementary functions of mathematics. A proof of this can ...
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44 views

Integral involving a Meijer-G function

I am having trouble with calculating the following integral: $$ \int_{0}^{\infty} \ln{(1 + \alpha x)\, G^{k,0}_{k,k}\left[e^{-x}\left|^{(a_k)}_{(b_k)} \right. \right]} \, dx, $$ where $\alpha > ...
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29 views

Correct integral to compute volume of a solid

I want to compute the volume obtained by rotating the bounded region of $y=3-x^2$, $y=2x$, $x \leq 0$ around the $y$-axis. I want to use the cylindrical shell method, so my integral is: ...
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55 views

Finding the length of an elliptical spiral

Okay, i had a very strange thought, it was "Is it possible to find the length of an elliptical spiral whose major and minor axes were decreasing?" Like for example lets say that $$ \frac{a}{b} = n ...
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34 views

Analysing an integral with its inverse integral [Image, and good explanation inside]

I have a function $f(x) = \dfrac{x}{\sqrt{1+x^2}}$ This means $f^{-1}(x) = (\pm)\dfrac{x}{1-x^2}$, where the negative solution is ignored for this problem. If I want to find a relation between the ...