0
votes
0answers
17 views

Integrals involving trigonometric functions and polynomes

Let $P(x)$ be a real polynome. Specify all such $P(x)$ that one of the next integrals converge: $$ \int_0^{\infty} sin(P(x))dx, \int_0^{\infty} cos(P(x))dx ? $$ Among special cases are such ...
1
vote
0answers
38 views

How naturally can functions defined by parametric integrals be interpolated from $\mathbb N$ to $\mathbb R^+$?

It is well known that several definite integrals $I_n$ containing a parameter $n\in\mathbb N$ can be expressed recursively (e.g. doing integration by parts) in terms of $I_{n-1} $ or $I_{n-2} $, and ...
0
votes
0answers
76 views

Integral of Bessel function of 1st kind with complex exponential

Does someone know the solution (simple closed form) of one of theses integrals: $$\int_0^t J_l(s) e^{-iA(t-s)}ds$$ $$\int_0^t \frac{J_l(s)}{s} e^{-iA(t-s)}ds$$ with $l>0$, $t>0$, $\Re(A)>0$, ...
9
votes
0answers
185 views

A multiple integral

Let us consider the multiple integral $$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots \int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots \cos ...
49
votes
2answers
2k views

Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?

I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here: Is it possible to express ...
8
votes
2answers
297 views

On the convexity of certain integrals involving Bessel functions

Let $n\geq 0$ be an integer and let $J_n=J_n(r)$ denote the usual Bessel function (of the first kind) of order $n$ i.e. one of the solutions to Bessel's differential equation ...
1
vote
1answer
87 views

Inversion of incomplete elliptic integral of third kind

I would like to know whether there is any solution available on the inversion of elliptic integrals of the third kind (incomplete)? That means that given $\Pi(n,u,m) = f(x)$, I would like to obtain ...
3
votes
2answers
344 views

How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration? $$ \int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0. $$ This post is related to my previous question here , ...
1
vote
0answers
113 views

Bound for a certain integral expression

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...
4
votes
2answers
110 views

Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?

The narrow Denjoy integral (which also goes by the names Henstock-Kurzweil integral, Perron integral, and Lusin integral) is a transfinite integration process defined by Denjoy in the early 20th ...
1
vote
1answer
132 views

Characterization of a particular integrable function

Let $f$ be a strictly positive function such that $\int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}xf(x)=1$ (i.e., a probability density function with expectation one). Let also $g$ be a nonnegative ...
54
votes
1answer
3k views

A hard integral identity on MATH.SE

The following identity on MATH.SE $$\int_0^{1}\arctan\left(\frac{\mathrm{arctanh}\ x-\arctan{x}}{\pi+\mathrm{arctanh}\ x-\arctan{x}}\right)\frac{dx}{x}=\frac{\pi}{8}\log\frac{\pi^2}{8}$$ seems to be ...
4
votes
1answer
315 views

Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...
5
votes
2answers
668 views

Integral with Dirac delta (me or wolfram mathematica?) [closed]

I asked the question on math.stackexchange but didn't get an answer so I came here. I tried to compute with Wolfram Mathematica the following integral $$I=\int_0^\pi\int_{-\infty}^\infty x e^{-\mathrm ...
1
vote
1answer
180 views

Convergence of a sum to the integral

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a 1-periodic function. I am looking about the conditions on $(a,b)\in\mathbb{R}^2$ such that we have the property : ...
11
votes
3answers
830 views

Applications of visual calculus

Mamikon's visual calculus (see Mamikon, Tom Apostol, Wikipedia) is a very beautiful and surprisingly efficient tool. The basis is Mamikon's theorem. The area of a tangent sweep is equal to the ...
14
votes
2answers
491 views

Evaluation of an $n$-dimensional integral

I asked the same question on math.se but got no answer there. Since it pertains to my current research, I decided to ask here: Let $n\in 2\mathbb{N}$ be an even number. I want to evaluate $$I_n := ...
1
vote
1answer
278 views

Why is there a formula for symbolic differentiation (chain and product rules) but not for symbolic integration? [duplicate]

Possible Duplicate: Why is differentiating mechanics and integration art? There is a formula for the derivative of any product, composite or sum of functions, in terms of the derivatives of ...
0
votes
0answers
591 views

Real analytic functions

I am quite confused with some ideas regarding the Real analytic functions. Just to introduce my questions: A function $f$ is real analytic on an open set $D$ of the real line if for any $x_0\in D$ ...
5
votes
3answers
3k views

Proof without words for surface area of a sphere [closed]

I love the book Proofs Without Words by Roger B. Nelsen. One of the proofs I liked the most was this: Area under one arch of a cycloid is 3 times the area of the wheel that traces it. You break the ...
15
votes
4answers
2k views

Is $x \, \tan(x)$ integrable in elementary functions?

I'm teaching Calculus and my students asked me to calculate the integral of $x \, \tan(x)$. I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be presented in ...
3
votes
2answers
334 views

Integration in several variables and elementary applications

This fall I'm teaching the "second half" of the standard entry-level undergraduate multivariable calculus course: the focus is on double and triple integrals, path integrals, Green's theorem, Stokes' ...
3
votes
3answers
294 views

Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $K\rightarrow \infty$?

What is the asymptotic behaviour or an upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b) \, dx$, for $a>b>0,$ as $K\rightarrow \infty$? Or any good reference for tools to tackle this question? ...
3
votes
1answer
538 views

Action Integral

In the theory of action-angle variables, you wind up having to solve integrals with a characteristic square-root behavior near the endpoints to express the action in terms of the orbital quantities. ...
1
vote
0answers
81 views

Semi implicit DAE integration using an implicit Runge Kutta scheme

I'm looking for some references regarding integration of DAEs in the form $M(t) \frac{dy}{dt} + G(y(t),t) + f(t) = 0, \quad y \in R^n, M(t) \in R^{nxn}$ using a high order implicit Runge Kutta ...
3
votes
1answer
306 views

Products of trigonometric functions with increasing frequencies

I am looking at weighted $L_2$ norms of a class of Littlewood polynomials, related to Walsh and Rademacher functions which made me look for pseudo-closed forms or computationally efficient expressions ...
2
votes
2answers
884 views

Definite Integral ∫_{0}^{∞} dx exp(−x^2−a exp(b x^2))

I've been trying without success to do $$\int_0^\infty dx\; \exp(-x^2) \exp(-a\exp(bx^2)).$$ It's not in my integral tables. Wolfram online integrator won't do it. It doesn't seem to be amenable to a ...
1
vote
2answers
437 views

High dimensional beta integral (a typo in Stein's book “singular integrals”)

Hello, When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake: $$ \int_{R^n} |x-y|^{-n+\alpha} ...
-1
votes
1answer
582 views

does equi-integrability implies uniform convergence?

A collection $\{f_n\}$ of real valued functions is said to be HK-equi-integrable on $I=[a,b]$, if there exists a gauge $\delta$ on $I$ such that for every $\epsilon>0$, there exists a $\delta$-fine ...
76
votes
5answers
6k views

Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference: $$ \frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2}, $$ where the ...
4
votes
4answers
2k views

Generalized Gauss-Green theorem

I am looking for a generalized version of the Gauss-Green theorem also known as the divergence theorem: http://en.wikipedia.org/wiki/Divergence_theorem A quick search on MathSciNet suggests that ...
72
votes
17answers
15k views

Why is differentiating mechanics and integration art?

It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by ...
1
vote
1answer
239 views

Integrate kˆ(n-1) / prod_{i=1…n} (kˆ(2)+x_iˆ{2}) dk between 0 and infinity, with x_i constants and n>=1? [closed]

[some formatting tweaked, and the question copied from the title to the main body, by YC] Hi, I've been struggling a lot to calculate this integral. $$ \int_0^\infty \frac{k^{n-1}}{\prod_{i=1}^n ...
30
votes
3answers
2k views

Is there a systematic method for differentiating under the integral sign?

This MO question by Tim Gowers reminded me of a question I've wondered about for some time. In the delightful book Surely You're Joking, Mr. Feynman!, Feynman praises the technique of differentiating ...
1
vote
2answers
485 views

limit of definite integral as $N \to \infty$

I'm interested in $\theta(N):=\int_0^1 (1-x)^{N-1} e^{xN} dx$. I'd like to show that $\theta(N)\sim c/\sqrt{N}$ as $N\to\infty$ and determine $c$. Any ideas?
16
votes
3answers
2k views

Weak and Strong Integration of vector-valued functions

This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference: Suppose $f:X\to E$ is a continuous function from a compact spaces (endowed ...
17
votes
6answers
2k views

Why not evaluate integrals using ODE-solvers?

Hello! I have a question about the relationship between numerical integration and the solution of ordinary differential equations (ODE). Suppose I want to evaluate the integral $I(x) = \int_{0}^{x} ...
5
votes
2answers
1k views

A tricky integral

Let $\alpha>0$ and $\beta\in\mathbb{R}$. I am looking for an explicit formula for the integral $$\int_{-\infty}^{\infty} (1+x^2)^{-1/2}e^{-\alpha x^2}e^{-i \beta x}dx.$$ I tried several changes ...
0
votes
1answer
905 views

Time scale calculus vs Lebesgue–Stieltjes calculus

About the same time, it seems, as I asked this question, a new post appeared on the wikipedia discussion page for Time scale calculus which suggests the Time scale derivative (aka Hilger derivative ...
1
vote
3answers
2k views

“Riemann–Stieltjes derivative”?

Can you define a "derivative" operator such that its antiderivative F(x) of f(x) can be used in the sense of F(b)-F(a) to calculate the Riemann–Stieltjes integral of f(x)? Perhaps it would be related ...
23
votes
3answers
3k views

What is the standard notation for a multiplicative integral?

If $f: [a,b] \to V$ is a (nice) function taking values in a vector space, one can define the definite integral $\int_a^b f(t)\ dt \in V$ as the limit of Riemann sums $\sum_{i=1}^n f(t_i^*) dt_i$, or ...
3
votes
1answer
1k views

Approximating a multiple sum with an integral

Hi, I want to approximate a multiple sum of the form $$\sum_{x_1+x_2+\cdots+x_m \leq n}e^{g(x_1,x_2,\ldots,x_m)},$$ where each $x_i$ is an integer between $0$ and $n$, by an integral ...
11
votes
4answers
6k views

Visualization of Riemann–Stieltjes Integrals

The Riemann–Stieltjes integral $\int_a^b f(x)\,dg(x)$ is a generalization of the Riemann integral. It is e.g. heavily used as a starting point for stochastic integration. The approximating ...
8
votes
4answers
2k views

An integral that somehow equals pi^2/6 and involves dilogarithms?

I am attempting to show that $$ \sum_{k \ge 1}^\infty {k^2 x^k \over (1+x^k)^2} \sim (1-x)^{-3} {\pi^2 \over 6} $$ as $x$ approaches 1 from below. The sum can be approximated by the integral $$ ...