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Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

113
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0answers
6k views

Is this really a categorical approach to integration?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical ...
31
votes
0answers
626 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider $f(x)=...
24
votes
0answers
1k views

Nested root integral $\int_0^1 \frac{dx}{\sqrt{x+\sqrt{x+\sqrt{x}}}}$

The bigger goal is to find the antiderivative: $$\int \frac{dx}{\sqrt{x+\sqrt{x+\sqrt{x}}}}~~~~~(*)$$ But I can settle for the definite integral in $(0,1)$. Motivation: $$\int \frac{dx}{\sqrt{x+\...
20
votes
0answers
2k views

A difficult integral

For $\gamma>0,\delta>0$, trying to evaluate this integral: $$ I=\int_0^H\frac{e^{i t x} \log\left(\frac{H}{H-x}\right) ^{\frac{1}{\gamma }-1} \left(\left(\frac{k}{H \log \left(\frac{H}{H-x}\...
15
votes
0answers
269 views

Geometric representation of Euler-Maclaurin Summation Formula

When reading Tom Apostol's expository article (or the free link), I was expecting more diagrams to come that follow the figure below (or this from the Wolfram project). It was a disappointment not ...
14
votes
0answers
597 views

Open problems in Federer's Geometric Measure Theory

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
14
votes
0answers
926 views

Integral involving Complete Elliptic Integral of the First Kind K(k)

I have run into an integral involving the complete elliptic integral, which can be put into the following form after changing integration variables to the modulus: $$\int_0^{\sqrt{\frac{\alpha}{1+\...
13
votes
0answers
439 views

Calculate using residues $\int_0^\infty\int_0^\infty{\cos\frac{\pi}2\Big(nx^2-\frac{y^2}n\Big)\cos\pi xy\over\cosh\pi x\cosh\pi y}dxdy,n\in\mathbb{N}$

Q: Is it possible to calculate the integral $$ \int\limits_0^\infty \int\limits_0^\infty\frac{\cos\frac{\pi}2 \left(nx^2-\frac{y^2}n\right)\cos \pi xy}{\cosh \pi x\cosh \pi y}dxdy,~n\in\mathbb{N}\...
13
votes
0answers
622 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x.$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
13
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0answers
626 views

Nontrivial trivial integrals

Consider two propositions in geometry: Circumscribe a right circular cylinder about a sphere. The surface area of the cylinder between any two planes orthogonal to the cylinder's axis equals the ...
12
votes
0answers
328 views

How to prove $\int_0^1x\ln^2(1+x)\ln(\frac{x^2}{1+x})\frac{dx}{1+x^2}$

How to prove$$\int_0^1x\ln^2(1+x)\ln\left(\frac{x^2}{1+x}\right)\frac{dx}{1+x^2}=-\frac{7}{32}\cdot\zeta{(3)}\ln2+\frac{3\pi^2}{128}\cdot\ln^22-\frac{1}{64}\cdot\ln^42-\frac{13\pi^4}{46080}$$ The ...
12
votes
0answers
6k views

Simpson's Rule for Double Integrals

Simpson's Rule for double integrals: $$\int_a^b\int_c^df(x,y) \,dx \,dy$$ is given by $$S_{mn}=\frac{(b-a)(d-c)}{9mn} \sum_{i,j=0,0}^{m,n} W_{i+1,j+1} f(x_i,y_j) $$ where: $$W= \begin{pmatrix} 1&...
12
votes
0answers
385 views

An integration to first order

I am having some trouble evaluating an integral -- involving taking an approximation. It would be great if someone could help me. I wish to evaluate $$\int_0^\pi {\cos\theta\cos \left[\omega t-{\...
12
votes
0answers
551 views

Finding a proper solution of a given functional

It's my first post here, but I worked very hard to find solution and I failed. Hereinafter, I skip physical background and directly proceed to my mathematical problem. No matter how, you know the ...
11
votes
0answers
279 views

Methodologies to Evaluate $\lim_{L\to \infty}\int_0^\infty \frac{\sin(Lx)}{x}\cos(x^3/3)\,dx$

In This Answer, I wrote "It is straightforward to show that $\displaystyle \lim_{L\to \infty}\int_0^\infty \frac{\sin(Lx)}{x}\,\cos(x^3/3)\,dx=\frac\pi2$." For completeness, I've included the "...
11
votes
0answers
356 views

A very useful lemma for Henstock-Stieltjes integration

I'd like to see a proof (or hints and outlines) for the following lemma, which is very useful to prove some interesting properties, including an Integration by Parts theorem for Henstock-Stieltjes ...
11
votes
0answers
577 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h \...
11
votes
0answers
446 views

Is this question solvable? $2$ non-linear equations and the proof that the solution is unique

As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
11
votes
0answers
1k views

How to compute this integral of Bessel functions?

I have $\alpha_\max$ a real number between $0$ and $\frac\pi2$. Furthermore $\zeta$ and $\xi$ are positive real numbers. Now I would like compute the integral $$\int_0^{\alpha_\max} \mathrm{e}^{i \...
11
votes
0answers
1k views

Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
10
votes
0answers
150 views

Geometric meaning of an integral on a compact Riemann surface

Let $X$ be a compact Riemann surface and fix a volume form $\Omega$ on $X$ such that $\int_X\Omega=1$. Now let's fix a function $g:U\subset X\to\mathbb R$ on $X$ with the following properties: $U=X\...
10
votes
0answers
134 views

Integral of $\int_0^{\infty} \ln\left|\frac{x+A}{x+B}\right|\frac{x}{e^{C x}\pm 1}dx$

so I have this integral to try and evaluate $$(*)=\int_0^{\infty} ln\left|\frac{x+A}{x+B}\right|\frac{x}{e^{C x}\pm1}dx$$ So far, I have managed to evaluate a very similar integral $$\int_0^{\infty}...
10
votes
0answers
120 views

How weird can the boundary be so that the fundamental theorems of vector calculus hold?

Let $\Omega$ be a connected open set in $\Bbb R^n$. Suppose that I want theorems in multivariables calculus like divergence theorem or its relative like Green's identities or even Stoke's theorem to ...
10
votes
0answers
466 views

Omega Constant Integral

Whilst reading this Math SE post, I saw that the OP mentioned the integral $$\int_0^\infty \frac{1+2\cos x+x\sin x}{1+2x\sin x +x^2}dx=\frac{\pi}{1+\Omega}$$ where $\Omega$ is the unique solution to ...
10
votes
0answers
1k views

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
10
votes
0answers
340 views

A generalization of an integral related with $\zeta(2)$

It is well-known that: $$ \int_{0}^{+\infty}\frac{x}{e^{x}-1}\,dx = \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} \tag{1}$$ but what is known about $$ I_2 = \int_{0}^{+\infty}\frac{x^2}{e^x-1-x}\,dx \...
10
votes
0answers
122 views

Evaluate the following integral involving $\sin \pi x$

Let $F: \Bbb{R} \to \Bbb{R}$ be defined by $$F(s)=\begin{cases}1, & \text{if }s\ge \dfrac12 \\[0.2cm]0, & \text{if }s< \dfrac12 \end{cases}$$ I need to evaluate $$\int^{1}_{0} F(\sin \pi ...
10
votes
0answers
1k views

Difficult integral for a marginal distribution

I am trying to derive a marginal probability distribution for $y$, and failed, having tried all methods to solve the following integral: $$p(y)=\int_0^{\frac{1}{\sqrt{2 \pi }}} \frac{\sqrt{\frac{2}{\...
10
votes
0answers
323 views

A difficult integral $\int_0^\infty \mathrm{d}t\frac{1}{t}\frac{1}{t-s-\mathrm{i}\epsilon}\frac{1}{X}\ln\frac{1-X}{1+X} $

Can anyone give any hints on how to rewrite this in terms of dilogarithms? $$ \int_{0}^{\infty}{{\rm d}t \over t}\,{1 \over t - s - {\rm i}\epsilon}\, {1 \over \,\sqrt{\, 1 - a/t\,}\,}\, \ln\left(1 - ...
10
votes
0answers
771 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2(\omega_{12}/2)\sin^2(\omega_{23}/2)d\omega_{23}d\omega_{12}$$ where: $f{} = 2\pi+2\tan^{-1}(y,x)$ $y = -...
9
votes
0answers
157 views

About the product of two Elliptic integrals

Let $z,x\in\left(0,1\right)$. It is possible to prove that $$\int_{0}^{1}\int_{0}^{1}\frac{1}{\sqrt{hy\left(1-h\right)\left(1-y\right)}}\frac{dydh}{\sqrt{\left(1+zhy\right)^{2}-4xzhy}}=\frac{4}{\pi^{2}...
9
votes
0answers
364 views

Why is backward Euler more stable?

I'm new to the idea of solving ODEs using the backward Euler. I have a system which I solve using the Backward Euler (actually backward Euler + Newton's method since I can't find a closed form ...
9
votes
0answers
230 views

Integration of a $k$-form over chains

In Spivak's Calculus on Manifolds, he defines the integral of a $k$-form over a $k$-chain, and proves a version of Stokes' theorem for this situation, before moving on to discuss the integral of a ...
9
votes
0answers
304 views

Interesting Integral with Parameters

I would like to compute the following integral: $$\int\frac{d^{2}\overrightarrow{q}}{2\pi}\int\frac{d^{2}\overrightarrow{k}}{2\pi}e^{i\overrightarrow{q}\cdot\overrightarrow{r}}\left(e^{i\...
9
votes
0answers
545 views

Change of variables for stochastic integral

Let $H$ be a previsible locally bounded process, and let $X$ be a continuous local martingale. If $T$ is a stopping time and $X^T=(X_{t+T}-X_{T},t\geq 0) $ then $$\int_T^{t+T}H_s.dX_s=\int_0^tH_{...
8
votes
0answers
125 views

Evaluating $\int_0^1 \frac{\mathrm dx}{(x^2+ax+1)^{n+1}}$ with real methods

I would like to know if there are any (preferably easier) methods of evaluating $$Q_n(a)=\int_0^1 \frac{\mathrm dx}{(x^2+ax+1)^{n+1}}$$ With real methods. Here's the way I did it. Complete the square:...
8
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0answers
168 views

What is $\int_0^1 \left(\tfrac{\pi}2\,_2F_1\big(\tfrac13,\tfrac23,1,\,k^2\big)\right)^3 dk$?

As in this post, define the ff: $$K(k)=K_2(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$ $$K_3(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac13,\tfrac23,1,\,k^2\right)}$$ $$K_4(k)={\...
8
votes
0answers
99 views

integrate $F(x)$: NO complex analysis, NO multivariable calculus

Suppose I have an elementary function $F(x)$ for which $\int_{-\infty}^\infty F(x) \, \text{d}x $ has an elementary value. Here 'elementary value' means anything generated by $0,1,+,-,\div,\times,\exp,...
8
votes
0answers
281 views

An $\operatorname{erfi}(x)e^{-x^2}$ integral

I want to find an elementary evaluation of $$I=\int_0^\infty \left(\frac{\sqrt\pi}2\operatorname{erfi}(x)e^{-x^2}-\frac1{1+2x}\right)dx$$ where $\operatorname{erfi}(x)=\frac{2}{\sqrt\pi}\int_0^...
8
votes
0answers
317 views

Show that $\int_{0}^{\pi\over 2}\arctan(\tan^8{(\pi^2{x}}))\mathrm dx={5\over 4}$

Can anyone help to provide a proof for $(1)$? Pleases! Thank you. $$\int_{0}^{\pi\over 2}\arctan(\tan^8{(\pi^2{x}}))\mathrm dx={5\over 4}\tag1$$ Enforcing $u=\tan^8{(\pi^2{x})}$ then $du={8\over \pi^...
8
votes
0answers
199 views

What are the mathematical foundations of the renormalisation group?

Briefly, RG refers to mathematical tools that allows systematic investigation of the changes of a physical system as viewed at different distance scales. These methods are very important in quantum ...
8
votes
0answers
358 views

Juantheron-like integral

While seeing this post, the following integral is just struck me \begin{equation} \int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1 \end{equation} I have tried like what user @OlivierOloa did in his ...
8
votes
0answers
248 views

Would an integral defined using partitions of an interval into infinitely many intervals make sense?

In the definition of Riemann integral or Darboux-integral we first study partitions (or tagged partition) of the given interval determined by finitely many points. To each partition and a function $f$ ...
8
votes
0answers
265 views

Seem to be encountering many interesting results from integrals in the form: $\int_0^\infty \frac{f(a,x)}{e^{2\pi x}-1}\text{d}x$

I have found interesting results from integrals in the form: $$I=\int_0^\infty \frac{f(a,x)}{e^{2\pi x}-1}\text{d}x$$ A few examples of interesting functions here are: $$f(a,x)=\sin(ax)\implies I=\...
8
votes
0answers
416 views

Stokes' Theorem and Vector Fields with Jump Discontinuities

What are the continuity requirements on a vector field $\boldsymbol{A}$ such that Stokes' theorem, $$ \iint_S\nabla\times\left[(\boldsymbol{\hat{x}}\cdot\nabla\phi)\boldsymbol{A}\right]\cdot d\...
8
votes
0answers
207 views

Can you add new functions to the set of elementary functions such that every function has an anti-derivative?

Its fairly well known that not every elementary function has an elementary anti-derivative. The common examples of this are $\exp(-x^2)$ and $\sin(x)/x$. The general workaround to this problem is to ...
8
votes
0answers
320 views

Help with the integral $\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$

Referring to a previous question, i want help with the integral : $$\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$$ Where $n,m$ ...
8
votes
0answers
116 views

Equivalence class of definite integrals

Let's assume we have a smooth function $f(x):[a,b]\to \mathbb{R}$ so that the integral $$\int_a^b f(x) dx$$ is finite. By performing various changes of variables, we can derive a large (infinite?) ...
8
votes
0answers
756 views

Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
8
votes
0answers
193 views

How to compute product integrals?

From the wikipedia article about product integrals I can see that if our function is scalar, then to compute type I product integral we can just take exponential of a usual integral: $$\prod_a^b f(x)^...