The value of $\underset{1}{\overset{a}{\int }}\left[x\right]f\text{'}\left(x\right)dx,a>1,$ where $\left[x\right]$ denotes the greatest integer not exceeding x is 

$$\begin{gathered} \text{ Let }\mathrm{I}={\int }_1^a \left[\mathrm{x}\right]{\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)\mathrm{dx},\mathrm{a}>1 \\ \text{ Let }\mathrm{a}=\mathrm{k}+\mathrm{h}\text{ where }\left[\mathrm{a}\right]=\mathrm{k},0\le \mathrm{h}<1 \end{gathered}$$
$$\begin{gathered} \begin{array}{r}\therefore {\int }_1^a \left[x\right]f^{\mathrm{\prime }}\left(x\right)dx \\ ={\int }_1^2 1f^{\mathrm{\prime }}\left(x\right)dx+{\int }_2^3 2f^{\mathrm{\prime }}\left(x\right)dx+\dots +{\int }_{k-1}^k (k-1)f^{\mathrm{\prime }}\left(x\right)dx+{\int }_k^{\mathrm{k}+\mathrm{h}} h{f}^{\mathrm{\prime }}\left(x\right)dx\\ \end{array} \\ \begin{array}{r}=\left(\mathrm{f}\right(2)-\mathrm{f}(1\left)\right)+2\left(\mathrm{f}\right(3)-\mathrm{f}(2\left)\right)+\dots +(\mathrm{k}-1)\left[\mathrm{f}\right(\mathrm{k})-\mathrm{f}(\mathrm{k}-1\left)\right]+\mathrm{k}\left(\mathrm{f}\right(\mathrm{k}+\mathrm{h})-\mathrm{f}(\mathrm{k}\left)\right)\\ \end{array} \end{gathered}$$
$\begin{array}{r}=-\mathrm{f}\left(1\right)-\mathrm{f}\left(2\right)-\mathrm{f}\left(3\right)\dots -\mathrm{f}\left(\mathrm{k}\right)+kf(\mathrm{k}+\mathrm{h})\\ =\left[\mathrm{a}\right]\mathrm{f}\left(\mathrm{a}\right)-\left\{\mathrm{f}\right(1)+\mathrm{f}(2)+\dots +\mathrm{f}(\left[\mathrm{a}\right]\left)\right\}\end{array}$