Tagged Questions

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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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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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, ...
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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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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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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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In the answer to this question Eric Näslund showed that logarithms can be written as the following limit of a sum: $$\displaystyle \log(x) = \lim_{k\to \infty } \, \sum\limits_{n=k}^{x k} \frac{1}{n}... 0answers 265 views why we cannot integrate on a nonorientable manifold? I feel it rather weird that there is a notion of integration when you glue a patch of paper to get a surface of cylinder while there is not a suitable notion when you glue it differently to get a ... 0answers 499 views Integral of two error functions (erf) In my research I came across the following integral: \int_{-\infty}^{+\infty}\frac{\partial{p(t)}}{\partial{t}}\frac{1}{4}\Big(1-\operatorname{erf}\Big(\frac{t-a}{\sigma\sqrt{2}}\Big)\... 0answers 942 views Do kernel functions of integral transforms have any special properties? From the Wikipedia page on integral transforms, it states that: ...an integral transform is any transform T of the following form:$$ (Tf)(u)=\int^{t_2}_{t_1}K(t,u)f(t)dt $$...There are ... 0answers 223 views Re: Rain droplets falling on a table This questions is almost exactly similiar to the the following question, with an extra condition : Rain droplets falling on a table Suppose you have a circular table of radius R. This table has ... 0answers 486 views Generalized Change of Variables Theorem? Is there a generalized form of the differentiable change of variables theorem for Lebesgue integrals? That is, if we consider the well known change of variables theorem: If \phi : X \rightarrow X is ... 0answers 174 views a integral of bivariate Gaussian random variables. I met the following problem when doing estimation and detection homework. The problem asks for a maximum likelihood estimator for (v,\rho) of bivariate joint Gaussian, where v is the common ... 0answers 221 views Orthogonal relation between divided functions. Let m and n are two integers such that , If m \neq n then$$\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 ...
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}^...