Intereting Posts

The group structure of elliptic curve over $\mathbb F_p$
Monos in $\mathsf{Mon}$ are injective homomorphisms.
The “architecture” of a finite group
Graph of symmetric linear map is closed
On the Whitney-Graustein theorem and the $h$-principle.
Isomorphic Group with $G=(\mathbb Z_{2^\infty}\oplus \frac{\mathbb Q}{\mathbb Z}\oplus \mathbb Q)\otimes_{\mathbb Z}\mathbb Q $
Banach space in functional analysis
Expected number of runs in a sequence of coin flips
How to evaluate $\int_{0}^1 {\cos(tx)\over \sqrt{1+x^2}}dx$?
Are infinitesimals dangerous?
If I remove the premise $a\neq b$ in this question, will the statement still be true?
what is the probability that the selected function maps prime numbers to prime numbers?
Weak convergence of a sequence of characteristic functions
Showing the metric $\rho=\frac{d}{d+1} $ induces the same toplogy as $d$
Evaluating $\sqrt{6+\sqrt{6+\cdots}}$

What better to start the year than a dazzling integral?

$$\int_{0}^{\infty}\left[1+\left(\frac{2013}{x+2013}+\cdots +\frac{2}{x+2}+\frac{1}{x+1}-x\right)^{2014}\,\right]^{-1}\,dx$$

Happy New Year to the mathematical community!

- Expressing the integral $\int_{0}^{1}\frac{\mathrm{d}x}{\sqrt{\left(1-x^3\right)\left(1-a^6x^3\right)}}$ in terms of elliptic integrals
- How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$
- Evaluate the definite integral $ \int_{-\infty}^{\infty} \frac{\cos(x)}{x^4 +1} \ \ dx $
- Closed form of $\ln^n \tan x\, dx$
- Improper integrals with singularities on the REAL AXIS (Complex Variable)
- Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$

(I am not too familiar with the posting policies on this site, hopefully this is not a major breach of rules)

- Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$
- Nice book on integrals
- Integrability of Thomae's Function on $$.
- Definitions of multiple Riemann integrals and boundedness
- Tables and histories of methods of finding $\int\sec x\,dx$?
- (complex measures) $d\nu=d\lambda +f\,dm \Rightarrow d|\nu|=d|\lambda| +|f|\,dm$ and $f\in L^1(\nu)\Rightarrow f\in L^1(|\nu|)$
- Principal value of the singular integral $\int_0^\pi \frac{\cos nt}{\cos t - \cos A} dt$
- $\text{Evaluate:} \lim_{b \to 1^+} \int_1^b \frac{dx}{\sqrt{x(x-1)(b-x)}}$
- Let $f:\to\mathbb{R}$ be a continuous function. Calculate $\lim\limits_{c\to 0^+} \int_{ca}^{cb}\frac{f(x)}{x}\,dx$
- Proof of the following fact: $f$ is integrable, $U(f,\mathcal{P})-L(f,\mathcal{P})<\varepsilon$ for any $\varepsilon>0$

The integral from $-\infty$ to $\infty$ is $$\frac{2\pi}{2014}\csc\left[\frac{\pi}{2014}\right]$$

See M.L. Glasser, *A remarkable property of definite integrals*, Math. Comp. **40**, 261 (1981).

Enfim, depois de passar alguns dias a fio tentando resolver esse desafio, acredito que a resposta numérica final seja $0{,}5 \cdot e \approx 1{,}35914$, onde $e \approx 2{,}71828$ é o número de Euler.

Anyway, after spending a few days on end trying to solve this challenge, I believe that the final numerical answer is $0{,}5 \cdot e \approx 1{,}35914$, where $e \approx 2{,}71828$ is the Euler number.

This is a numerical approximation, not the exact result. The function being integrated relies only on the sum $n / (x + n)$, where $n$ is an integer and belongs to the interval $[1, 2013]$. The sum has a single root in the interval $[0, \infty)$. That is, $x ≈ 938{,}17268…$ the value of the sum is zero. Well, then the function being integrated is equal to $1$. Therefore, we can easily see that the function being integrated in the interval $[0; 938{,}17268…)$ is growing. And in the interval $(938{,}17268; \infty)$ is decreasing.

Using the method of trapezoids in the vicinity of $x \approx 938{,}17268$ we can determine an approximation for the numerical integration. As the function values are dwindling, there is no significant change in the first 5 digits houses. So I suggested that $0{,}5 \cdot e \approx 1{,}35914$ is an approximation to the result. Furthermore, it is possible to prove this result using Maple with interactive integration algorithm.

It is important to remember that the function is not continuous the full extent of the real numbers. How can there be integral from $-\infty$ to $\infty$, mentioned in the first reply?

- 2013 USAMO problem 5
- Proof by induction that $ 169 \mid 3^{3n+3}-26n-27$
- Is an integer a sum of two rational squares iff it is a sum of two integer squares?
- Behavior of Gamma Distribution over time
- Is every noninvertible matrix a zero divisor?
- Groups with 20 Sylow subgroups
- Principal Bundles, Chern Classes, and Abelian Instantons
- What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)
- About phi function
- Minimum radius of N congruent circles on a sphere, placed optimally, such that the sphere is covered by the circles?
- Solving Recurrence equation
- Finding the sum of series $\sum_{n=0}^∞ \frac{2^n + 3^n}{6^n}$
- Principal period of $\sin\frac{3x}{4}+\cos\frac{2x}{5}$
- Remainder term in Taylor's theorem
- Decimal Expansion of Pi