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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270 views

existence of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
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4k views

The negative integral meaning

Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? Does it simly mean that the said area is under the the x - axis, ...
4
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270 views

On an application of the Abel-Plana formula

Referring to a previous question, i am having a hard time trying to do the integral: $$f(s)=-i\int_{0}^{\infty}\frac{\log \left[1+\frac{\left(s\log(1+ix) \right )^{2}}{4\pi ^{2}} \right ]-\log \...
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100 views

Evaluate $\int \dfrac {x^2}{\sqrt{\arctan x}} dx$

Is there any closed form expression of $$\int \dfrac {x^2}{\sqrt{\arctan x}} dx?$$
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464 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $\mathscr{C}^1$ on $(c,d)$. Then ...
4
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112 views

Convection-diffusion-reaction problem

I seek to solve to the system $$ \frac{\partial \phi_{a}}{\partial t} = D_{a} \frac{\partial^{2} \phi_{a}}{\partial x^{2}} - v_{a} \frac{\partial \phi_{a}}{\partial x} + \mathfrak{K}_{b}\phi_{b} ...
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86 views

Evaluation of the double integral $\iint_{[0,1] \times [0,1]}{\left(\frac{xy(1-x)(1-y)}{2-xy}\right)^n \frac{dxdy}{2-xy}}$

How to prove that $$\large \lim_{n \to \infty} \left(\iint_{[0,1] \times [0,1]}{\left(\frac{xy(1-x)(1-y)}{2-xy}\right)^n \cdot\frac{1}{2-xy}\; \text{d}x\text{d}y}\right)^{\LARGE\frac{1}{n}}=\max_{(x,...
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190 views

Integral of product spherical harmonics

I'm trying to calculate this integral: $$\int_0 ^\pi\int_0^{2\pi} [\sin\theta(\cos\theta \sin \phi + \sin\theta \cos(2\phi))]^2 \sin\theta d\phi d\theta $$ We could compute this integral directly....
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231 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := \...
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207 views

Integral $ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx,\quad \alpha,n \in \mathbb{R}^{0+}. $$ This integral is old. I am also looking for literature on these ...
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282 views

The Monster PolyLog Integral $\int_0^\infty \frac{Li_n(-\sigma x)Li_m(-\omega x^2)}{x^3}dx$

I am trying to solve this integral $$ \int_{0}^{\infty} {{\rm Li}_{n}\left(-\sigma x\right){\rm Li}_m\left(-\omega x^{2}\right) \over x^{3}}\,{\rm d}x $$ which is from some high school training ...
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195 views

Integral $=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ I_{n,m}(\alpha,\sigma,\omega,r)=\int_0^...
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80 views

How can I integrate this zeta function expression?

Can you integrate this function: $$f(k)=\exp\left(-\Re\left(\sum\limits_{n=1}^{n=scale} \frac{1}{n} \zeta(1/2+i \cdot k)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot k-1)}}\right)\right)$$ with ...
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114 views

Symbolic math engines barf on this ostensibly tractable integral.

$$\frac14 \int_{-M\pi}^{N\pi - s} \cos(tu/M) \cos((t+s)u/M)(1-\cos(t/M))(1-\cos((t+s)/N))\space \mathrm d t$$ with integer $u$. Alpha runs out of time. Maxima gives a tremendous result that can ...
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110 views

Tricky Gaussian; a good method?

Can anyone help evaluate $$ \int_{0}^{r}\int_{0}^{r+\epsilon-\sqrt{r^{2}-x^{2}}}e^{-\beta\left(x^{2}+y^{2}\right)} \, dy \, dx $$ ($r$ and $\epsilon$ are constants)?
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194 views

$\int e^{-x} \log \log x dx$ - another special integral?

I came across this integral in some old notes. After several unsuccessful attempts I ran it in WA and got an interesting result: the antiderivative (closed form) doesn't exist, but the bounded ...
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212 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
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193 views

How to integrate $\left(1+\ln(x)\right)\sqrt{1+(x\ln(x))^2}$ with Risch algorithm?

How would you integrate $\left(1 + \ln\left(x\right)\right)\, \sqrt{1 + \left(x\ln\left(x\right)\right)^{2}\,}$ using the Risch algorithm? I want to know this because Mathematica is using the Risch ...
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107 views

The Leibniz rule in Euler's works

Does anyone know if the Leibniz rule (the method of differentiation under the integral sign), or a variation thereof, has ever appeared in any of Euler's papers? Any references would be appreciated. ...
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110 views

Integrals for children

I was just reviewing a chapter on integration defined through step functions, and was wondering how would you explain the concept of an integral of a step function to a child ?
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91 views

Doubt about computability of integrals over open sets.

In Spivak's Calculus on Manifolds after showing that partitions of unity exists, Spivak defines integrals of functions over open sets as follows. He says: "An open cover $\mathcal{O}$ of an open ...
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89 views

Computing volume of ball in $n$ dimensions

Let $B^n(a)$ denote the closed ball of radius $a$ in $\mathbb{R}^n$, centered at $0$. Show that $v(B^n(a))=\lambda_n a^n$ for some constant $\lambda_n$, where $v(X)$ denotes the volume of $X$. By ...
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163 views

Approximate the integral $\int_0^\pi \sin(x^3)\mathrm{d}x$ with a standard pocket calculator

I came over the following integral $$ \int_0^\pi \sin(x^3) \mathrm{d}x $$ when a friend of mine tried to approximate it. The most obvious way is to use taylors formula, and then turn the integral ...
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177 views

Limiting behavior of an integral involving incomplete Gamma function

I am wondering about the limiting behavior as $k\rightarrow\infty$ of the following integral: $$I(k)=\frac{2^{-k/2}}{\Gamma(k/2)}\int_{f(k)}^\infty t^{k/2-1}e^{-t/2}\left(\frac{\gamma\left(\frac{k}{2}...
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65 views

Can Fredholm integral equation of the first type be represented as a differential equation?

Can Fredholm integral equation of the first type be represented as a differential equation? In other words, given a Fredholm integral equation of the second type does there exist a differential ...
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56 views

integration of simple function

So I have this question I saw this week which I do not understand the answer for it. Let $a,b \in \mathbb{R} a <b \ $ and let $f_1:[a,b] \rightarrow \mathbb{R}$ and I know that $\int_a^x \mathrm{...
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538 views

Riemann-Stieltjes Integral computation (step function?)

Im trying to integrate this, using theorem 7.9 of apostol's book: $$\int^{10}_0 f(x)d\alpha(x) $$ $f(x) = x^2$ and $\alpha(x)= 3\chi(7,9](x)$ Where $\chi(x)$ is $0$ everywhere except $1$ in the ...
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138 views

Taking an integral of e^(another integral) with respect to the limit of integration of (another integral)

How would I go around integrating $$\int_0^\infty \left( \exp{\left(-\int_{k_0}^{k_0+t} (k_1 + k_2s)(k_3+k_4s)^{s} ds \right) } \right) dt \text{,}$$ where $k_i$ are constants? Is it solvable ...
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101 views

There exists unique $g$ s.t. $g(x) = f(x) + A\int_0^1\sin(x-y)g(y)dy$

I'm doing past papers for a first course on functional analysis. We are not allowed to assume any results from real analysis or topology, so I was surprised to find an exam question, where I couldn't ...
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259 views

Solve a nasty integral

Can anyone help me with this very nasty integral: $$\int \frac{e^{-\frac{x^2}{2}}~x\left(-1+2b^2+2x^2\right)}{\sqrt{1-e^{x^2}} \sqrt{b^2+x^2}} dx,$$ where $b\in\mathbb{R}$. I've tried pretty much ...
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316 views

Stokes' Theorem problem

Let $M \subset \mathbf{R}^n$ be oriented compact smooth $k$-manifold and $\alpha$ be a $C^1$ diferential $(k-1)$-form defined in a neighborhood of M. Use Stokes' theorem to prove that \begin{align*} ...
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97 views

Stokes' Theorem and Measure Zero Sets

This is probably a very naive question but I am trying to connect two pieces of information in my head regarding integration of differential forms and integration with respect to a measure. The first ...
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515 views

Bounds for the exponential integral

In Abramowitz and Stegun: Handbook of Mathematical Functions (on page 229, property 5.1.20) it is found that $$ \frac{1}{2} \log \left(1 + \frac{2}{x} \right) < \exp(x) E_1(x) < \log \left(1 + ...
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173 views

How can I Create an integral that can only be evaluated via complex contour integration?

In Richard Feynman's book, Surely You're Joking Mr. Feynman!, he says: One time I boasted, "I can do by other methods any integral anybody else needs contour integration to do." So Paul [Olum]...
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451 views

An absolutely continuous cumulative distribution function that fails to have a Riemann-integrable pdf.

We know that if a real-valued random variable $ X $ on a probability space has an absolutely continuous cumulative distribution function (cdf) $ F $, then $ X $ possesses a probability density ...
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244 views

difficult integral involving arcsin(x)

I have a difficult integral to compute. I know the result by guessing the answer, but need to know the method of calculation. The integral is $$ \int_{a}^{b}{\rm d}p\,{p \over p^{2} - 2\mu}\, \...
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178 views

Simplifying an integral arising in Physical Chemistry

I am struggling to understand the following transition (encountered in a paper on Physical Chemistry). Let $$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int w(R)\exp[-tw(...
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137 views

Gaussian Integral with non-polynomial exponent

I am currently trying to evaluate this Integral: $$\int\limits_{u_0}^{u_1} \exp\left[-\angle(H(u),N)^2\right]du$$ Where $H(u)=:\overrightarrow{a}+(\overrightarrow{b}-\overrightarrow{a})*u-\...
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81 views

Dominated convergence on $e^{-n^2 t} t^{s/2-1}$

I am trying to apply the Dominated Convergence Theorem to show that $$\sum_{n\ge 1} \int_0^1 e^{-n^2 t} t^{s/2-1}dt= \int_0^1 \sum_{n\ge 1}e^{-n^2 t} t^{s/2-1}dt$$ as soon as $s>1$. I've noted ...
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335 views

Heaviside unit step- and delta function

The following question is right from the book: Show that $$ H(x-x_i) = \int_{-\infty}^x \delta(x_0-x_i)dx_0\, $$ satisfies $$ H(x-x_i) \equiv \begin{cases} 0 & x < x_i \\ 1 &...
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718 views

The Lebesgue Criterion for Riemann Integrability — a proof without using the concept of oscillation.

I am trying to prove the Lebesgue Criterion for Riemann Integrability without using the concept of oscillation. The Lebesgue Criterion for Riemann Integrability states that if $ f: [a,b] \to \mathbb{...
4
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0answers
163 views

When are sums and integrals “identical” in form?

In the answer to this question Eric Näslund showed that logarithms can be written as the following limit of a sum: $$\displaystyle \log(x) = \lim_{k\to \infty } \, \sum\limits_{n=k}^{x k} \frac{1}{n}$...
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265 views

why we cannot integrate on a nonorientable manifold?

I feel it rather weird that there is a notion of integration when you glue a patch of paper to get a surface of cylinder while there is not a suitable notion when you glue it differently to get a ...
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499 views

Integral of two error functions (erf)

In my research I came across the following integral: \begin{equation} \int_{-\infty}^{+\infty}\frac{\partial{p(t)}}{\partial{t}}\frac{1}{4}\Big(1-\operatorname{erf}\Big(\frac{t-a}{\sigma\sqrt{2}}\Big)\...
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942 views

Do kernel functions of integral transforms have any special properties?

From the Wikipedia page on integral transforms, it states that: ...an integral transform is any transform $T$ of the following form: $$ (Tf)(u)=\int^{t_2}_{t_1}K(t,u)f(t)dt $$ ...There are ...
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223 views

Re: Rain droplets falling on a table

This questions is almost exactly similiar to the the following question, with an extra condition : Rain droplets falling on a table Suppose you have a circular table of radius R. This table has ...
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0answers
486 views

Generalized Change of Variables Theorem?

Is there a generalized form of the differentiable change of variables theorem for Lebesgue integrals? That is, if we consider the well known change of variables theorem: If $\phi : X \rightarrow X$ is ...
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174 views

a integral of bivariate Gaussian random variables.

I met the following problem when doing estimation and detection homework. The problem asks for a maximum likelihood estimator for (v,$\rho$) of bivariate joint Gaussian, where v is the common ...
4
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0answers
221 views

Orthogonal relation between divided functions.

Let $m$ and $n$ are two integers such that , If $m \neq n$ then $$\int _a^b {\dfrac{f_n(x)}{f_m(x)} \ dx}=0$$ If $m=n$ then $$\int _a^b {\dfrac{f_n(x)}{f_n(x)} \ dx}=\int _a^b 1 \ dx=b-a$$ ...
4
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0answers
117 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 $f:\mathbb{R}^...