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 ...

learn more… | top users | synonyms (3)

5
votes
1answer
881 views

Change of variables for a Dirac delta function

I have often seen the following equality in Physics textbooks. $$\int_{\mathbb{R}}\delta\left(\alpha x\right)f\left(\alpha x\right)|\alpha|dx=\int_{\mathbb{R}}\delta(u)f(u)du$$ or ...
5
votes
1answer
209 views

Integral with hyperbolic cosine squared

Does anyone can give me a hint how to integrate the following: $$\int_0^\infty{\frac{x^2 {\rm d}x}{\mathrm{cosh}^2(x)}}.$$ The answer is $\frac{\pi^2}{12}$ (taken from the book). I've started with ...
5
votes
1answer
549 views

Is integration a continuous functional on the Skorohod space?

I have read several times that integration is a continuous functional on the Skorohod space $D[0,1]$, i.e., the set of all cadlag functions on $[0,1]$ equipped with the Skorohod metric; in symbols, ...
5
votes
2answers
375 views

Math Courses involving clever integration techniques

I am a third year undergraduate mathematics student. I learned some basic techniques for simplifying sums in high school algebra, but I have encountered some of the more interesting techniques in my ...
5
votes
1answer
328 views

Definite integral of tetration between $0$ and $1$

In my old writes I found next formula, where is ${_{}^2}x$ is tetration: $$\int_0^1 {_{}^2}x \ dx = \sum\limits_{i=1}^\infty \frac {(-1)^{i+1}} {{_{}^2}i} \approx 0.783430511\ldots$$ And now I am ...
5
votes
2answers
2k views

Why does the definition of an integral specify a closed interval?

Here's the definition of an integral from Wikipedia: Given a function $f$ of a real variable $x$ and an interval $[a, b]$ of the real line, the definite integral $$\int_a^b f(x) \, dx$$ ...
5
votes
3answers
164 views

A proof problem from a first time real analysis course

Given a continuous function $g:[a,b]\to\Bbb R$, if there exists a number $K>0$ s.t. for all $x\in[a,b]$, $|g(x)| \le K \int_a^x |g|$, prove $g(x)=0$ for all $x\in[a,b]$. And I tried to derive some ...
5
votes
2answers
386 views

Variation of $f(x)=x^\eta\sin^\varepsilon(\frac{1}{x})$

I'm asked to characterize the values of the parameters $\eta, \varepsilon$ for which the above function is of bounded variation on $[0,1]$, when we set $f(0)=0$. By "bounded variation", I mean that ...
5
votes
3answers
1k views

How can I find the area of the shadow?

Consider a lit candle placed on a cylinder. If the candle is placed at the center of the top surface, let the distance from the origin (center of the surface) to the end of the shadow be $r$. In this ...
5
votes
1answer
189 views

Evaluate $\int_1^\infty \cosh^{-1}(x) \ln(x^2-1) \exp \left(- \frac{x}{T} \right) dx $

I would be interested in any clue on how to evaluate the following integral $$\int_1^\infty \cosh^{-1}(x) \ln(x^2-1) \exp \left(- \frac{x}{T} \right) dx $$ I have tried integration by parts but it ...
5
votes
1answer
153 views

Double integral $\int_{z=u}^{+\infty}\int_{t=u}^{+\infty}\frac{e^{-Az}}{z+B}\frac{te^{-tD}}{t-zC}\,dtdz$

I am doing research, and while calculating a closed form expression, I got a form of integration like the following: ...
5
votes
1answer
348 views

Average sine of an angle between two rays in a cone

I'm looking for an average value of sine of an angle between two rays, lying within a cone with a certain angle. Given a cone with an aperture of ${2\chi}$ and two rays lying within the cone. The ...
5
votes
1answer
1k views

How to find this integral?

I am trying to find the integral $$\oint_c Re(z)\;dz$$ where c is a circle $$|z|=2$$ I don't know what to do. I tried some things but I don't know if I am correct. $e^{i\theta} = \cos \theta +i ...
5
votes
1answer
69 views

Continuous functions question

I am stuck on the problem: Find all continuous functions $h$ satisfying $$\int_{0}^{x}h(y)dy=\left [ h(x) \right ]^{2}+C$$ for some constant $C$.
5
votes
1answer
249 views

Dyson series and T product (II)

After reading the previous posts related to the Dyson series, I have decided to open a new thread because there is something that I am still not understanding. It concerns the expression: $$ ...
5
votes
1answer
647 views

vector calculus - Integral over vector

Our physics prof wrote the following equation: $\int\frac{\vec{r}}{r^3}d\vec{r} = \int\frac{1}{r^2}dr$ This is logical as long as I argue that $\vec{r}$ and $d\vec{r}$ are parallel, which is why the ...
5
votes
1answer
402 views

$2\cdot\int_0^\infty \frac{a-u^2}{\left( u^2+\frac{a^2}{b-a}\right) \left(u^2+\frac{b^2}{b-a} \right) \sqrt{\cdots } }\mathrm {d}u $

at the moment I am trying to reproduce the results of a paper. There, it turns out that a specific physical problem is mapped onto an integral to be calculated: $$I(\Theta; a, b) = ...
5
votes
2answers
110 views

How do I get $ \int_0^1 \frac{dz}{\sqrt{z(z - 1\,)(z+1\,)}} = \frac{\sqrt{\pi}}{2} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{9}{4})}$?

While reading physics papers I found a very interesting integral so I decided to write it down. Let $p(z) = z^ 3 - 3\Lambda^ 2 z$ where $\Lambda$ could be any number. If you want $\Lambda = 1$ and ...
5
votes
1answer
158 views

Verify integration of $ \int\frac{\sqrt{2-x-x^2}}{x^2}dx $

This is exercise 6.25.40 from Tom Apostol's Calculus I. I would like to ask someone to verify my solution, the result I got differs from the one provided in the book. Evaluate the following integral: ...
5
votes
1answer
98 views

Help integrating the transition probability of the Brownian Motion density function.

1. Problem: Given the Brownian Motion with Drift: $$ dx = \mu \, dt+\sigma \, dW $$ It can be shown that the transition density function is the following: $$ p(x, t) = \frac{e^{-\frac{(x-\mu ...
5
votes
1answer
35 views

Why does 'allowing y to move more' change the value of this integral?

Consider the integral: $$ \int_{-1}^1\int_{|y|}^1(x+y)^2dxdy $$ The domain of integration is the triangle described by $|y|\leq x\leq 1$. I've drawn this domain of integration and thought that if ...
5
votes
1answer
67 views

Two similar integration about continued fractions

Prove that \begin{align*} \int_0^{+\infty} \cfrac{\sin nx}{x + \cfrac{1}{x + \cfrac{2}{x + \cfrac{3}{x + \cdots}}}} \, dx &= \cfrac{\sqrt {\cfrac{\pi }{2}} }{n + \cfrac{1}{n + \cfrac{2}{n + ...
5
votes
2answers
96 views

Why is it legal to take the antiderivative of both sides of an equation?

first, I must apologize for somewhat misleading a title. To save both your and my time, I will go straight to the point. By definition, an indefinite integral, or a primitive, or an antiderivative ...
5
votes
1answer
76 views

Laplace Transform

Suppose that $F(s)=L[f(t)]$ and $G(s)=L[g(t)]$, where $L$ is the Laplace transformation $$F(s)=L[f(t)]=\int_0^{+\infty}e^{-st}f(t)dt.$$ I'm trying to prove that: $$\textrm{If}\ \ \lim_{t\to 0^+} ...
5
votes
1answer
228 views

Problems with ln(ax) equations.

After fiddling around with the ln() function, I arrived at a problem. I have found that $a \approx 1.39095$. However, I couldn't find the exact value. Using the Lambert w function, I have already ...
5
votes
1answer
45 views

$\sup_{x\in \mathbb{R}} \int_0^\varepsilon e^{ay}|f(x-y)|dy\to 0$ when $\varepsilon$ goes to $0$?

Let $a$ be a real number and $f:\mathbb{R}\to \mathbb{R}$ a function. If the function $f$ is bounded, then we can see that $$\sup_{x\in \mathbb{R}} \int_0^\varepsilon e^{ay}|f(x-y)|dy\to0,$$ as ...
5
votes
2answers
123 views

Computing an improper integral with respect to a parameter

I am motivated by this problem.Let us compute an improper integral with respect to a parameter:$$F(x)=\int_{1}^{\infty}\frac{e^{-xy}-1}{y^{3}}dy,\quad x\in[0,\infty).$$ The following is my ...
5
votes
1answer
81 views

Evaluate integral with gaussian curvature

I thought evaluating it in the following way: $$\begin{align} \int_0^{2\pi}\int_0^{\pi}K(x,y)\sqrt{\det(g_{ij})} \, dy\,dx &= \int_0^{2\pi}\int_0^\pi \sqrt{\det L_{ij}}\cdot \sqrt{{\frac{\det ...
5
votes
2answers
68 views

A problem related to integration in $L^1$

If $f\in L^1[0, 1]$ and $\int_{0}^1 x^nf(x)=0$ for all $n = 0,1,2,...$then prove that $f$ is identically zero almost everywhere. This would be very easier to prove if $f$ were continuous on $[0, 1]$ ...
5
votes
1answer
221 views

$f(x) = \sin x $, if $x$ is rational and $f(x) = x$, if $x$ is irrational.

A function $f$ is defined on $[0,\pi/2]$ by $$f(x) = \begin{cases} \sin x, & x\ \text{is rational},\\ x, & x\ \text{is irrational} \end{cases}$$ Then to find the upper integral and the ...
5
votes
5answers
80 views

Properties of $L^2(-1,1)$ functions

I want to show that there is no function $v \in L^2(-1,1)$ with $\int_{-1}^{1} v(x)\phi(x) dx = 2\phi(0)$ for all $\phi \in C^\infty_0(-1, 1)$ ($\phi$ is $0$ everywhere but $[-1,1] $). I know about ...
5
votes
1answer
71 views

A product version of Riemann integral

Motivated by Riemann sum in Riemann integral and motivated by relations between infinite series and infinite products we ask: Assume that $f:[0, 1]\to \mathbb{R}$ is a positive function. Assume ...
5
votes
1answer
126 views

Integration by parts for general measure?

Let $\mu$ be a general measure, suppose $f,g$ has compact support on $\mathbb{R}$, when does the integration by parts formula hold $$\int f'g d\mu = - \int g'fd\mu?$$ I know in general this is false, ...
5
votes
1answer
93 views

Does $\int_1^\infty\sin (\frac{\sin x}{x})\mathrm d x$diverge or not?

Does $\int_1^\infty\sin (\frac{\sin x}{x})\mathrm d x$diverge or not? If it converges, does it converge conditionally or absolutely? I guess that it converges conditionally, also,I think it may be ...
5
votes
1answer
130 views

Consider the equation: $x' = f(t,x)$. Prove that there is a two-way correspondence between the initial and the limits of the solutions.

Consider the equation: $$x' = f(t,x)$$ wherein, $$|f(t,x)| \leq \phi(t)x, \forall(t,x) \in \mathbb{R}\times \mathbb{R} $$ $$ \int^{\infty}_a\phi(t)\,dt< \infty$$ where $a \in \mathbb{R}$. If in ...
5
votes
2answers
78 views

Conceptual question on substitution in integration [duplicate]

In calculus we learn about the substitution method of integrals, but I haven't been able to prove that it works. I mainly don't see how manipulations of differentials is justified, i.e how $dy/dx = ...
5
votes
1answer
127 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 $$ ...
5
votes
1answer
117 views

Computation of a indefinite integral: $ \int \frac {dx} {x^n-1}$

This trouble arose when I earlier played with W|A, I found that it can compute $\int \frac{dx}{x^{11}-1}$, which has a huge messy answer. With great uncertainty, this's my work so far: Let ...
5
votes
1answer
508 views

Riemann Integrability in $\Bbb R^2$

Define the General Subdivision $S$ of a rectangle $R$ in $\Bbb R^2$ as a collection $E_1,...,E_k$ of Jordan regions such that none of them has interior points in common, and: $$R \subset ...
5
votes
1answer
82 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 > ...
5
votes
1answer
136 views

Estimating $\int_0^x f(x-t)f'(t)dt$

I'm attempting to estimate $\int_0^x f(x-t)f'(t)dt$ in terms of a simple asymptotic expression with an error term for some 'well-behaved' functions, namely $f = O(x)$, of class $C^1$ or higher, with ...
5
votes
2answers
136 views

Integrate a periodic absolute value function [duplicate]

\begin{equation} \int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor \end{equation} I ...
5
votes
1answer
78 views

How find this limit $\lim_{p\to 0^{+}}\left(\int_{a}^{m-p}f(x)dx+\int_{m+p}^{b}f(x)dx\right)$

Give real numbers $a,b$ such that $0<a<b$ and $m=\dfrac{a+b}{2}<\dfrac{\pi}{4}$,Evaluate $$\lim_{p\to 0^{+}}\left(\int_{a}^{m-p}f(x)dx+\int_{m+p}^{b}f(x)dx\right)$$ where ...
5
votes
1answer
76 views

Integral (close form?)

I'm struggling to evaluate the following integral: $\displaystyle \int_{-1}^{1}\frac{1+2x^2+3x^4+4x^6+5x^8+6x^{10}+7x^{12}}{\sqrt{\left ( 1+x^2 \right )\left ( 1+x^4 \right )\left ( 1+x^6 \right ...
5
votes
1answer
105 views

Integrating over a power of the infinitesimal

I don't know if the title makes sense (or if the question makes sense at all for that matter) but here I go. Suppose I have a piecewise constant function $y=f(x)$ with $x,y\in\mathbb{R}^+$, described ...
5
votes
1answer
102 views

Sorting out some integrals from physics

I'm doing some physics for a change, and I'm trying to sort things out a bit. From the definitions of mass, torque, momentum and angular momentum I've come up with the following integrals: ...
5
votes
2answers
78 views

An inequality with $a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}}$

Let the sequence $(a_n)_n$ defined by $$a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}}$$ 1)Prove that $$\frac12 \leq a_n \leq ...
5
votes
2answers
185 views

Integral with rational functions of powers and exponentials

Any ideas how to solve: \begin{equation} \int_0^\infty x^{n+\frac{1}{2}} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t} dx \end{equation} where $a$ and $t$ are real, positive constants; $n$ is a positive ...
5
votes
1answer
733 views

How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
5
votes
1answer
222 views

How to find this integral $\int_{0}^{1}\ln\ln\bigl(1/x+\sqrt{(1/x^2)-1}\,\bigr)dx$ [duplicate]

How do I compute this integral ? $$I=\int_{0}^{1}\ln{\left(\ln{\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)}\right)}dx$$ In the math chatroom someone suggests setting ...