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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29 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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43 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?
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41 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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61 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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23 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 ...
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41 views

Monotone convergence, measure-theory, is this excercise correct?

Here is the exercise: I have some questions: Is this correct when k starts with 1?, the Taylor series with e starts with 0? But does the zero disappear in some way?, I can not see how. I know that ...
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36 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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29 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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54 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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52 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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24 views

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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36 views

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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122 views

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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32 views

Numerical Integration of Highly Oscillatory Integral with Misbehaving Derivatives

I'm attempting to numerically handle an equation of the following form: \begin{equation*}f: x \rightarrow \int_{0.00001}^{2} d\omega e^{i \omega x} f(\omega)\end{equation*} where $f(\omega) = ...
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37 views

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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46 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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71 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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57 views

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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37 views

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 : ...
2
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45 views

Measure inequality implies integral inequality?

Let $f$ and $g$ be non-negative, integrable functions on a measure space with measure $\mu$, and suppose there is some constant $c > 0$ such that for every $t \geq 0$, the inequality $\mu(\{f \geq ...
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38 views

Region of triple integral

It's very hard to me to visualize the shape of this region, does anyone could help me, thanks!
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65 views

Second order nonlinear ordinary differential equation. Help please

Can someone help me with this differential equation $$ay''(t)y(t)+2y'(t)=\left(b+\frac{c}{t^2}\right)y(t)^2$$
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57 views

Integrate $(\cos(x)-e^{-x^2})/x$

How does one integrate the following? $$\int_{0}^{\infty}\frac{\cos(x)-e^{-x^2}}{x} dx$$ Is it possible to do this without using mellintransformation?
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59 views

Prove that $\int_{0}^{1}{f(x)dx}=0$ for non negative function such that the continuity point are zeros of $f$.

Let $f:[0,1] \to \mathbb R$ be a non negative function such that if $c \in [0,1]$ is a continuity point of $f$ then $f(c)=0$. Prove that $\int_{0}^{1}{f(x)dx}=0$. This problem it's about Riemann ...
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76 views

Fallacy in integration technique to find a sphere's surface area?

The problem is to find the surface area of a sphere. Consider the first approach as shown in the following picture (the approach is used in Wikipedia too): In this first approach, the sphere is ...
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120 views

Question on the Fourier Transform, specifically concerning polynomials

Suppose $P$ is a polynomial of degree $\ge 2$ with distinct roots, none lying on the real axis. Calculate: $$\int_{- \infty}^{\infty}\frac{e^{-2 \pi i x \xi}}{P(x)}dx,\space \space \space\xi \in ...
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53 views

Prove double integral inequality

Prove the following statement: If $f(x,y)$ and $g(x,y)$ are integrable over a close region $R$ and $f(x,y)≥g(x,y)$, then: $$\iint[f(x,y) \ d(x,y)]≥\iint[g(x,y) \ d(x,y)]$$ I am stuck on this question. ...
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81 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
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59 views

Invariant functions under integral transforms

We all know Fourier transform has invariants such as $e^{-x^2}$, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering ...
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118 views

Indefinite Integral

I tried to solve this indefinite integral $$\int\frac{1}{1+\tan^{-1}x}dx$$ I try taking the change of variable $u=\tan^{-1}x$ but I fail to reach a solution. Can anyone help me. Thanks in advance.
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Using Polars to Approximate a Cartesian line: Approximating an Integral

I have the equation of the lower semicircle of radius $r$ centred at a distance $a+r$ above the x-axis $$ f(x)=r+a-\sqrt{r^{2}-x^{2}} $$ which I can approximate (for small $x$) as $$ f(x)\approx ...
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55 views

Prove an equation about fractional integral

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Can anyone help me solve this? I really appreciate. If $f$ is continuous on $[0, \infty)$, for $\alpha \gt ...
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33 views

Calculus: Reduction formula

For this question, I can find out $I3$, but I have no idea how to find the reduction formula. Please advise me.
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47 views

Question about integral

I want to prove the inversion theorem on $R^{k}$.Then I need to compute the integral : $\int_{R^{1}}\exp^{-\sqrt{x^{2}+a^{2}}+itx}dx$ I have no ideal how to deal with it.I will appreciate your help. ...
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42 views

How to estimate this special integral?

Let $\theta\in(0,\pi)$, and $$ {\rm I}\left(\lambda,\theta\right) = \int_{\theta}^{2\pi - \theta} \left[({1 \over \sin\left(t\right)}\frac{\partial}{\partial t})^{k}{\rm e}^{\left({\rm i}\lambda - ...
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25 views

Stationary Phase approximtion for this type of integral

I would like to approximate the following integral for large $t$ : $I(t)=\int_0^{\pi}dx f(x)e^{iS(x)t}$ $S$ is real and $S'(0)=S'(\pi)=0$. $f$ is real and $f(0)=f(\pi)=0$ and $f'(0)=f'(\pi)=0$. ...
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32 views

Proof of a Proposition

I am having trouble trying to proof a proposition that appears in a paper. It begins with ...
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108 views

Attempted proof of the first part of the Fundamental Theorem of Calculus

So I was trying to prove the first part of the Fundamental Theorem of Calculus in a different way using the Riemann sum definition of the definite integral, rather than the way it was presented in my ...
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92 views

Transforming a Riemann-Stieltjes integral related to the factorial

I have been able to show that $$\log n! = \int_{1 + \epsilon}^n \log x \, d\lfloor x \rfloor$$ but I have not been successful trying to transform this Riemann-Stieltjes integral to an ordinary ...
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69 views

Name of theorem?

I am trying to understand a proof which uses the following statement without further explanation, so I am wondering if this is a well known theorem? For the unit ball $B$ with radius $r>0$ and the ...
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92 views

Is it possible to switch limit from inside to outside of integral in this case?

Let $C$ be an open connected subset of $\mathbb{C}$. Let $f:[a,b]\times C \rightarrow \mathbb{C}$ be a function. Assume $f(-,z):[a,b]\rightarrow \mathbb{C}$ is continuous and $f(t,-):C\rightarrow ...
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42 views

Question about finding volume using integration?

The question is: Find the volume of the solid whose base is a circle $x^2 + y^2 = 81$ and the cross sections perpendicular to the $x-axis$ are triangles whose height and base are equal. Now what the ...
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39 views

Need help evaluating $\lim\limits_{n \to \infty} \frac{1}{n} \int_1^n \Vert\frac{n}{x}\Vert dx$

$$ \mbox{Evaluate}\quad \lim_{n \to \infty}{1 \over n}\int_{1}^{n}\left\Vert\,n \over x\,\right\Vert \,{\rm d}x $$ Where $\left\vert\left\vert\, x\,\right\vert\right\vert : \mathbb{R} \to \mathbb{R}$ ...
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75 views

Integrating With Respect To $x$

Suppose I have the first derivative of the function $y$, $\displaystyle \frac{dy}{dx} = g(x)$. Furthermore, suppose I want to obtain the function $y$ by integrating with respect to $x$: ...
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127 views

integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
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44 views

calculating sum of a limit of integral

I am trying to calculate the following expression $$ \sum_{m=0}^{\infty} \frac{1}{m!} \lim_{n \to \infty} \int_{\{(x,y):2x^2+y^2<n^2 \}}\left( 1 - \frac{2x^2+y^2}{n^2}\right)^{n^2} x^{2m}dx~dy ...
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42 views

comparison of two integrals

Let $n \in N$. How to compare two integrals: $$ I_1=\int_0^{\infty}\left(\frac{\sin t}{t}\right)^n dt \quad \text{and} \quad I_2=\int_0^{\pi}\left(\frac{\sin t}{t}\right)^n dt\,\, ? $$ I've beet ...
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31 views

Integral computation - what's going on?

Let $\lambda >0$ and denote $$ \lim _{\varepsilon \to 0+} \frac{1}{|\xi |^2 - (\lambda + i\varepsilon )^2} = \frac{1}{|\xi |^2 - (\lambda + i0)^2}. ...
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59 views

Area between two curves with a certain domain.

I am trying to find the area between two curves over a certain domain. The region of integration is between $xy=5$, $x=9-y^2$ and the lines $y=1$ and $y=2$. I have to show that this can be written as ...
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39 views

Compute an indefinite integral of logarithms.

Is there a simple way of proving the following identity: \begin{eqnarray} \int \log(x) \log(x^2 + (x + W)^2) dx = \\ 2 x + \left(1 - \log(x)\right) \left(\log(e^{-\pi/2} W) W + 2 x + W ...