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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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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76 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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80 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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158 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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63 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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54 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 ...
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494 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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137 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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96 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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227 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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270 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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93 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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445 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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166 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 ...
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417 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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224 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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135 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 ...
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80 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 ...
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320 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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587 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 ...
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157 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} ...
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230 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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824 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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220 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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462 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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170 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 ...
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218 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 $$\int _a^b {\dfrac{f_n(x)}{f_n(x)} \ dx}=\int _a^b 1 \ dx=b-a$$ ...
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110 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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283 views

Computing complex principal value integral - sgn-function?

I currently face a less appealing integral which emerged computing the expectation of some random variable. It reads as (omitting all unnecessary constants except $\alpha\in(0,1)$) $$ PV ...
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Riesz-Type Representation Theorems for Convex Functionals

It is well known that any positive linear functional $L$ on the spase $C_c([a,b])$ of functions continuous on an interval $[a,b]$ with compact support can be written as \begin{align*} ...
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80 views

tough definite integral: $\int_0^\frac{\pi}{2}x\ln^2(\sin x)~dx$

Any ideas on $\int_0^\frac{\pi}{2}x\ln^2(\sin x)\ dx$ ? Best numerical approximation I can get is $0.2796245358$ Is there even a closed form solution?
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123 views

Approach to this integral

According to a standard literature $$\frac{1}{\sqrt{2\pi}q}\int^{-d}_{−∞} \sum ^∞_{k=1} \frac{(A^2/2q^2)^k}{2^k(k!^2)} e^\frac{−r^2}{2q^2}He_{2k}(\frac{r}{q})dr=\frac{1}{\sqrt{2\pi}} \sum ^∞_{k=1} ...
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63 views

How to integrate exponential and power function?

I am trying to solve the following integral $$\int_{0}^{\infty}e^{-(ax+bx^c)}\,dx ; ~~~a,b,c>0.$$ I tried using partial functions but that didn't lead to anything. Any suggestion?
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44 views

Airy transform of gaussian on half-line: $\int_{0}^\infty dx\, e^{-x^2}\text{Ai}(y-x)$

Background. The Airy transform of $f$ is defined as $$\int_{-\infty}^\infty dx\, f(x)\,\text{Ai}(y-x)\;.$$ $\text{Ai}$ denotes Airy function, $$\text{Ai}(x)=\frac{1}{\pi}\int_{-\infty}^\infty ...
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47 views

Prove the real-version of Riemann–Lebesgue lemma

I've been told to prove the real-version of Riemann–Lebesgue lemma, which is: for $f$ integrable and $2\pi$ periodic: $$ \lim_{n\to\infty} \int_{0}^{2\pi} f(x)\cos(nx) \ dx = \lim_{n\to\infty} ...
3
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248 views

Evaluating $\int_{0}^{\infty} \left[\left(\frac{2015}{2015+x}+\cdots +\frac{2}{2+x}+\frac{1}{1+x}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x$

I need to evaluate $$\int_{0}^{\infty} \left[\left(\frac{2015}{2015+x}+\cdots +\frac{2}{2+x}+\frac{1}{1+x}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x $$ I've been told that the way forward is ...
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42 views

Computing the limit of this integral,

This is Part 6 (last part) of a problem statement of an old comprehensive exam question that I am working on. It asks to evaluate $$\lim_{r_0 \to 0} \int_{-\infty}^{\infty} ...
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134 views

Is $h(\mathbf{a},b)=\int_\Omega f(\mathbf{x})e^{-g(\mathbf{x})}\mathrm{d}\mathbf{x}$ differentiable?

Let $f\colon\Bbb{R}^n\to\Bbb{R}$ be an affine function and $g\colon\Bbb{R}^n\to\Bbb{R}$ be a non-negative function. We define $h\colon\Bbb{R}^n\times\Bbb{R}\to\Bbb{R}$ as follows $$ ...
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28 views

Green's Theorem with respect to a given polar region.

Using Green's Theorem, compute the counterclockwise circulation $I$ of $\vec{F}=\langle-\sqrt{x^2+y^2},\sqrt{x^2+y^2}\rangle$ around the region defined by the polar coordinate inequalities $7 ...
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86 views

Convergence for a improper integral $\int^b_a fg$

Let $f$ be continuous on [a,b) such that $\int^b_a f$ converges. If $g'$ is locally integrable and has a constant sign on [a,b), prove that $\int^b_a fg$ converges. Edit: We can assume that the limit ...
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62 views

How to evaluate: $\int_0^1x^{n-1}(1-x)^{n+1}dx$

How can I evaluate the following integral? ($n \in R$, $n>0$) $$\int_0^1x^{n-1}(1-x)^{n+1}dx$$ I was solving the following problem (as practice) in school: Prove that the sum of $n+1$ terms of ...
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What is the number of zeros of antiderivatives of $(x-1)(x-2)^2(x-3)^3(x-4)^4$?

For each $x \in \mathbb{R}$, let $f(x) = (x-1)(x-2)^2(x-3)^3(x-4)^4$. This defines a function $f : \mathbb{R} \to \mathbb{R}$. There is a unique natural number $k$ such that every antiderivative of ...
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47 views

Calculate $\int_0^{2\pi}\int_0^b \frac{r}{a-r\cos \vartheta}dr\,d\vartheta$.

$$\int_0^{2\pi}\int_0^b \frac{r}{a-r\cos \vartheta}dr\,d\vartheta$$ When I put this in Mathematica just give me a long formula, but I know that ...
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23 views

Complex line integration with assumptions

Let $f: \mathbb{C} \to \mathbb{C} $ be a holomorphic function with $$ \lim_{\lvert z \rvert\to\infty} \frac{f(z)}{z^{n-1}} = 0$$ for some $n\in\mathbb{N}$. How can I prove that $$ \lim_{r\to\infty} ...
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41 views

Minimum number of circles to completely cover a sphere

I've encountered a problem which I have some idea of solving but am befuddled on how to proceed. Here is the full problem: Suppose that you wish to cover a 1-km radius, spherical planet with ...
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72 views

Contour Integral of Square root Function. Branch Cuts

I am doing a physics problem and have come across a contour integral that I just don't know how to solve. I do not have the complex analysis background and I am wondering if anyone can explain how to ...
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46 views

Evaluate $\int\frac{1}{x}\coth(ax)\sin^2(\frac{xt}{2})dx$

I am trying to integrate this function: $\int_0^\infty\frac{1}{x}\coth(\frac{\hbar x}{2kT})\sin^2(\frac{xt}{2})dx$ which Wolframalpha (for me) returns nothing, just a blank screen. I thought that it ...
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48 views

Solving an integral within an integral

I have been trying for a while now to solve the following integral: $$ \int_0^t e^{-ct} \left(\int_0^t(1-M_r t)^a e^{ct} dt\right) (1-M_r t)^{-(a+1)} dt $$ I know the answer is: $$ \frac {1}{M_r^2} ...
3
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57 views

Measurable function in $L^2$-norm

Let $f$ be a measurable function. Assume you know that $$\sup_{||g||_2=1} \left|\int fg\,\right|$$ exists. Does this mean that $$\sup_{||g||_2=1} \left|\int fg\,\right| = ||f||_2?$$
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$Q=A\times B$. if $\int_Q f$ exists, then $\int_{y\in B}f(x,y)$ exists for $x\in A-D$, where $D$ is a set of measure zero in $\mathbb{R^k}$.

Let $A$ be a rectangle in $\mathbb{R^k}$; let $B$ be a rectangle in $\mathbb{R^n}$; let $Q=A\times B$. Let $f: Q\to \mathbb{R}$ be a bounded function. Show that if $\int_Q f$ exists, then ...