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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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 $$\int_a^b f(x) \, dx$$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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: ...
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 ...
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. $e^{i\theta} = \cos \theta +i ... 1answer 69 views Continuous functions question I am stuck on the problem: Find all continuous functions$h$satisfying $$\int_{0}^{x}h(y)dy=\left [ h(x) \right ]^{2}+C$$ for some constant$C$. 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: $$... 1answer 647 views vector calculus - Integral over vector Our physics prof wrote the following equation: \int\frac{\vec{r}}{r^3}d\vec{r} = \int\frac{1}{r^2}dr This is logical as long as I argue that \vec{r} and d\vec{r} are parallel, which is why the ... 1answer 402 views 2\cdot\int_0^\infty \frac{a-u^2}{\left( u^2+\frac{a^2}{b-a}\right) \left(u^2+\frac{b^2}{b-a} \right) \sqrt{\cdots } }\mathrm {d}u at the moment I am trying to reproduce the results of a paper. There, it turns out that a specific physical problem is mapped onto an integral to be calculated:$$I(\Theta; a, b) = ... 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$and ... 1answer 158 views Verify integration of$ \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: ... 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: $$p(x, t) = \frac{e^{-\frac{(x-\mu ... 1answer 35 views Why does 'allowing y to move more' change the value of this integral? Consider the integral:$$ \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 ... 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 + ... 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 ... 1answer 76 views Laplace Transform Suppose that F(s)=L[f(t)] and G(s)=L[g(t)], where L is the Laplace transformationF(s)=L[f(t)]=\int_0^{+\infty}e^{-st}f(t)dt.$$I'm trying to prove that:$$\textrm{If}\ \ \lim_{t\to 0^+} ... 1answer 228 views Problems with ln(ax) equations. After fiddling around with the ln() function, I arrived at a problem. I have found thata \approx 1.39095$. However, I couldn't find the exact value. Using the Lambert w function, I have already ... 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$fis bounded, then we can see that $$\sup_{x\in \mathbb{R}} \int_0^\varepsilon e^{ay}|f(x-y)|dy\to0,$$ as ... 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 ... 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 ... 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] ... 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] byf(x) = \begin{cases} \sin x, & x\ \text{is rational},\\ x, & x\ \text{is irrational} \end{cases}$$Then to find the upper integral and the ... 5answers 80 views Properties of L^2(-1,1) functions I want to show that there is no function v \in L^2(-1,1) with \int_{-1}^{1} v(x)\phi(x) dx = 2\phi(0) for all \phi \in C^\infty_0(-1, 1) (\phi is 0 everywhere but [-1,1] ). I know about ... 1answer 71 views A product version of Riemann integral Motivated by Riemann sum in Riemann integral and motivated by relations between infinite series and infinite products we ask: Assume that f:[0, 1]\to \mathbb{R} is a positive function. Assume ... 1answer 126 views Integration by parts for general measure? Let \mu be a general measure, suppose f,g has compact support on \mathbb{R}, when does the integration by parts formula hold$$\int f'g d\mu = - \int g'fd\mu?$$I know in general this is false, ... 1answer 93 views Does \int_1^\infty\sin (\frac{\sin x}{x})\mathrm d xdiverge or not? Does \int_1^\infty\sin (\frac{\sin x}{x})\mathrm d xdiverge or not? If it converges, does it converge conditionally or absolutely? I guess that it converges conditionally, also,I think it may be ... 1answer 130 views Consider the equation: x' = f(t,x). Prove that there is a two-way correspondence between the initial and the limits of the solutions. Consider the equation:$$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$$where a \in \mathbb{R}. If in ... 2answers 78 views Conceptual question on substitution in integration [duplicate] In calculus we learn about the substitution method of integrals, but I haven't been able to prove that it works. I mainly don't see how manipulations of differentials is justified, i.e how dy/dx = ... 1answer 127 views Klein-Gordon field commutator integral? Consider a Klein-Gordon field \phi, which satisfies$$(\Box+ \omega_0^2)\phi=0$$on points x \equiv \{x_0,\vec{x}\},y\equiv \{y_0,\vec{y} \} of 4D Minkowski-spacetime. The field commutator is$$ ... 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}$, which has a huge messy answer. With great uncertainty, this's my work so far: Let ... 1answer 508 views Riemann Integrability in$\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: $$R \subset ... 1answer 82 views Integral involving a Meijer-G function I am having trouble with calculating the following integral:$$ \int_{0}^{\infty} \ln{(1 + \alpha x)\, G^{k,0}_{k,k}\left[e^{-x}\left|^{(a_k)}_{(b_k)} \right. \right]} \, dx, $$where \alpha > ... 1answer 136 views Estimating \int_0^x f(x-t)f'(t)dt I'm attempting to estimate \int_0^x f(x-t)f'(t)dt in terms of a simple asymptotic expression with an error term for some 'well-behaved' functions, namely f = O(x), of class C^1 or higher, with ... 2answers 136 views Integrate a periodic absolute value function [duplicate] $$\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$$ I ... 1answer 78 views How find this limit \lim_{p\to 0^{+}}\left(\int_{a}^{m-p}f(x)dx+\int_{m+p}^{b}f(x)dx\right) Give real numbers a,b such that 0<a<b and m=\dfrac{a+b}{2}<\dfrac{\pi}{4},Evaluate$$\lim_{p\to 0^{+}}\left(\int_{a}^{m-p}f(x)dx+\int_{m+p}^{b}f(x)dx\right)$$where ... 1answer 76 views Integral (close form?) I'm struggling to evaluate the following integral: \displaystyle \int_{-1}^{1}\frac{1+2x^2+3x^4+4x^6+5x^8+6x^{10}+7x^{12}}{\sqrt{\left ( 1+x^2 \right )\left ( 1+x^4 \right )\left ( 1+x^6 \right ... 1answer 105 views Integrating over a power of the infinitesimal I don't know if the title makes sense (or if the question makes sense at all for that matter) but here I go. Suppose I have a piecewise constant function y=f(x) with x,y\in\mathbb{R}^+, described ... 1answer 102 views Sorting out some integrals from physics I'm doing some physics for a change, and I'm trying to sort things out a bit. From the definitions of mass, torque, momentum and angular momentum I've come up with the following integrals: ... 2answers 78 views An inequality with a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}} Let the sequence (a_n)_n defined by$$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 ... 2answers 185 views Integral with rational functions of powers and exponentials Any ideas how to solve: $$\int_0^\infty x^{n+\frac{1}{2}} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t} dx$$ where$a$and$t$are real, positive constants;$n$is a positive ... 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. 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 ...