Evaluating $\int_{0}^{\infty} \left[\left(\frac{2015}{2015+x}+\cdots +\frac{2}{2+x}+\frac{1}{1+x}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x$

Solution 1:

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$\ds{\int_{0}^{\infty}\bracks{\pars{{2015 \over 2015 + x} + \cdots + {2 \over 2 + x} + {1 \over 1 + x} - x}^{2016} + 1}^{-1}\,\dd x:\ {\large ?}}$.

\begin{align} &\int_{0}^{\infty}\bracks{\pars{{2015 \over 2015 + x} + \cdots + {2 \over 2 + x} + {1 \over 1 + x} - x}^{2016} + 1}^{-1}\,\dd x \\[5mm] = &\ \int_{0}^{\infty}{\dd x \over \mc{F}^{2016}\pars{x} + 1}\quad \mbox{where}\quad\mc{F}\pars{x} \equiv \sum_{k = 1}^{2015}{k \over k + x} - x \end{align}

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\begin{align} \mc{F}\pars{x} & \equiv \sum_{k = 1}^{2015}{k \over k + x} - x = 2015 - x\sum_{k = 1}^{2015}{1 \over k + x} - x \\[5mm] & = 2015 - x\sum_{k = 1}^{\infty}\pars{{1 \over k + x} - {1 \over k + 2015 + x}} - x \\[5mm] & = 2015 - x\pars{H_{x + 2015} - H_{x}} - x\qquad \pars{~H_{z}:\ Harmonic\ Number~} \end{align}

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\begin{align} &\bbx{\mc{F}\pars{x} = 2015 - \pars{\vphantom{\large A}H_{x + 2015} - H_{x} + 1}x} \\[5mm] &\mbox{Some characteristic behaviours of}\ \mc{F}\pars{x}\ \mbox{are}\ \left\{\begin{array}{l} \ds{\mc{F}\pars{0} = 2015} \\[2mm] \ds{\mc{F}\pars{x} \to -\infty\quad \mbox{as}\quad x \to \infty} \\[2mm] \ds{\mc{F}'\pars{x} \leq 0\,,\quad \forall\ x \geq 0} \\[2mm] \ds{\mc{F}\pars{r} = 0\,,\quad r \approx 939.105} \\[2mm] \ds{\mc{F}\pars{939.105} \approx -6.28337 \times 10^{-4}} \\[2mm] \ds{\mc{F}'\pars{939.105} \approx -1.46404} \end{array}\right. \\[1cm] &\mbox{Hereafter, I'll perform a numerical evaluation which is based in the Laplace Method}: \\ &\int_{0}^{\infty}{\dd x \over \mc{F}^{2016}\pars{x} + 1} = \int_{0}^{r}{\dd x \over \mc{F}^{2016}\pars{r - x} + 1} + \int_{0}^{\infty}{\dd x \over \mc{F}^{2016}\pars{x + r} + 1} \\[5mm] \approx &\ 2\int_{0}^{\infty} \expo{-\bracks{\mc{F}'\pars{r}}^{\large 2016}x^{\large 2016}}\,\dd x = {2 \over \verts{\mc{F}'\pars{r}}}\int_{0}^{\infty}\expo{-x^{\large 2016}}\,\dd x \\[5mm] = &\ {2 \over \verts{\mc{F}'\pars{r}}}\,{1 \over 2016} \int_{0}^{\infty}x^{1/2016 - 1}\expo{-x}\,\dd x = {1 \over 1008}\,{1 \over \verts{\mc{F}'\pars{r}}}\,\Gamma\pars{1 \over 2016} \\[5mm] & \approx \bbx{1.36569} \quad\mbox{with}\quad \left.\mc{F}'\pars{r}\right\vert_{\ r\ \approx\ 939.105} \approx -1.46404 \end{align}

This value $\ds{\pars{~1.36569~}}$ is the numerical one reported by $\texttt{@achille hui}$ in another answer comment.

Solution 2:

Edited: This integral evaluation is of a related integral which is equivalent to $\int^{\infty}_0(x^{2016}+1)^{-1}dx$, for the original integral in the question, differs in evaluation for limits from $0$ to $\infty$ owing to symmetry considerations discussed in the comments.

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$$I=\frac{1}{2}\int_{-\infty}^{\infty} \left[\left(\frac{2015}{2015+x}+\cdots +\frac{2}{2+x}+\frac{1}{1+x}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x $$ $$I=\frac{1}{2}\int_{-\infty}^{\infty} \left[\left(\sum^{2015}_{i=1}\frac{i}{x+i}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x $$

Now, letting $f(x)=\frac{1}{x^{2016}+1}$, and noting that $f(x)=f(-x)$, $$I=\frac{1}{2}\int_{-\infty}^{\infty} f\left(\sum^{2015}_{i=1}\frac{i}{x+i}-x\right)\mathrm{d}x=\frac{1}{2}\int_{-\infty}^{\infty} f\left(-\left(\sum^{2015}_{i=1}\frac{i}{x+i}-x\right)\right)\mathrm{d}x$$ $$I=\frac{1}{2}\int_{-\infty}^{\infty} f\left(x-\sum^{2015}_{i=1}\frac{i}{x-(-i)}\right)\mathrm{d}x \tag {1}$$ Using Glasser's Master Theorem, $$I=\frac{1}{2}\int^{\infty}_{-\infty} f(x)\ \mathrm{d}x=\frac{1}{2}\int^{\infty}_{-\infty} \frac{1}{x^{2016}+1}\ \mathrm{d}x=\int^{\infty}_{0} \frac{1}{x^{2016}+1}\ \mathrm{d}x \tag {2}$$

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Now we know that $$B(a,b)=\int^{\infty}_0\frac{t^{a-1}}{(1+t)^{a+b}}dt$$ From $(2)$,after substituting $x^{2016} =t$, $$I=\frac{1}{2016}\int^{\infty}_{0}\frac{t^{\frac{1}{2016}-1}}{(1+t)^{\frac{1}{2016}+\frac{2015}{2016}}}dt=\frac{1}{2016}B(\frac{1}{2016},\frac{2015}{2016})$$ Therefore $$\color{red}{I=\frac{1}{2016}\frac{\Gamma(\frac{1}{2016})\Gamma(\frac{2015}{2016})}{\Gamma(1)}=\frac{\pi}{2016\sin(\frac{\pi}{2016})}\approx1.0000004047320180811575}$$