Can the value of this integration be found? $$\displaystyle\int_{0}^{1}\dfrac{\displaystyle\mathrm{d}x}{ \sum_{k = 1}^{n}\sqrt{x+k}}$$
1) For $k=0$, $\displaystyle \int_{0}^{1}\frac{1} {\sqrt{x}}\, \mathrm dx=2$
2) for $k=1$, $$\int_{0}^{2}\frac{1}{\sqrt{x}+\sqrt{x+1}}\, \mathrm dx=\int_{0}^{1}(\sqrt{x+1}-\sqrt{x} )\, \mathrm dx=\frac{2}{3}(2^{\frac{3}{2}}-1)$$
3) for $k=2$, $$\int_{0}^{1}\frac{1}{\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}}\, \mathrm dx =\int_{0}^{1}\frac{(\sqrt{x}+\sqrt{x+1})-\sqrt{x+2}}{x-1+2\sqrt{x(x+1)}}\, \mathrm dx \\ =\int_{0}^{1}\frac{(\sqrt{x}+\sqrt{x+1}-\sqrt{x+2})(x-1-2\sqrt{x+1})}{x^2-2x-5}\, \mathrm dx \\ =-\int_{0}^{1}\frac{\sqrt{x}(x-1-2\sqrt{x+1})}{2\sqrt{6}(x-1+\sqrt{6})}\, \mathrm dx -\int_{0}^{1}\frac{\sqrt{x+1}(x-1-2\sqrt{x+1})}{2\sqrt{6}(x-1+\sqrt{6})}\, \mathrm dx -\int_{0}^{1}\frac{\sqrt{x+2}(2\sqrt{x+1}+1-x)}{2\sqrt{6}(x-1+\sqrt{6})} \, \mathrm dx +\int_{0}^{1}\frac{\sqrt{x}(x-1+2\sqrt{x+1})}{2\sqrt{6}(x-1+\sqrt{6})} \, \mathrm dx +\int_{0}^{1}\frac{\sqrt{x+1}(x-1+2\sqrt{x+1})}{2\sqrt{6}(x-1+\sqrt{6})}\, \mathrm dx +\int_{0}^{1}\frac{\sqrt{x+2}(2\sqrt{x+1}+1-x)}{2\sqrt{6}(x-1-\sqrt{6})} \, \mathrm dx \\ =(0.197824376869)+(0.1404434996429)+(0.251789392006)+(0.378195288745)+(-0.493066563246)+(-0.325023194397) \\ =0.1501627965 $$
So for $k\geq3$, how much is its value, and by finding the integral value in terms of $n$ can Its value be found for any value of $k$?
Waiting for your help.
I am quite skaptical about a possible closed form but I think that we could have some reasonable approximations.
Around $x=0$, we have $$\sqrt{x+k}=\sqrt{k}+\frac{x}{2 \sqrt{k}}-\frac{x^2}{8 k^{3/2}}+\frac{x^3}{16 k^{5/2}}-\frac{5 x^4}{128 k^{7/2}}+O\left(x^5\right)$$ $$\sum_{k=1}^n\sqrt{x+k}=H_n^{\left(-\frac{1}{2}\right)}+\frac{ H_n^{\left(\frac{1}{2}\right)}}{2} x -\frac{H_n^{\left(\frac{3}{2}\right)}}{8} x^2 +\frac{H_n^{\left(\frac{5}{2}\right)}}{16} x^3 -\frac{5H_n^{\left(\frac{7}{2}\right)}}{128} x^4 +\cdots$$
Making the long division and integrating does not make much problems.
To illustrate, I used the first expansion up to $O(x^{11})$ and below are the results for a few $n$ $$\left( \begin{array}{ccc} n & \text{approximation} & \text{exact} \\ 1 & 0.8358802795 & 0.8284271248 \\ 2 & 0.3599023214 & 0.3595321155 \\ 3 & 0.2151816216 & 0.2151109818 \\ 4 & 0.1477755342 & 0.1477517722 \\ 5 & 0.1097547559 & 0.1097440975 \\ 6 & 0.0857627226 & 0.0857570600 \\ 7 & 0.0694528161 & 0.0694494560 \\ 8 & 0.0577585089 & 0.0577563539 \\ 9 & 0.0490304149 & 0.0490289510 \\ 10 & 0.042308743 & 0.0423077034 \end{array} \right)$$