A problem for the New Year

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!

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

Solutions Collecting From Web of "A problem for the New Year"

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?