$\underset{0}{\overset{5}{\int }}\left[x\right]dx=$

$\begin{array}{l}G.I={\int }_0^1 0dx+{\int }_1^2 1dx+{\int }_2^3 2dx+{\int }_3^4 3dx+{\int }_4^5 4dx\\ =0+(x{)}_1^2+2(x{)}_2^3+3(x{)}_3^4+4(x{)}_4^5\\ =1+2\left(1\right)+3\left(1\right)+4\left(1\right)=10\end{array}$