Question

Let ${\int }_0^2[x^2-x+1]dx$; where $[.]$ denotes the greatest integer function be equal to $\frac{k-\sqrt{m}}{n}$.Find $k+m+n$ ?

Answer 1

We know $[x^2-x+1],\forall xϵ[0,2]$ could be plotted as shown in figure (a).
$\therefore [x^2-x+1]$ is as shown in figure (b)
$\therefore I={\int }_0^2[x^2-x+1]dx$
$={\int }_0^{1}0dx+{\int }_1^{\frac{(1+\sqrt{5})}{2}}1dx+{\int }_{\frac{(1+\sqrt{5})}{2}}^{2}2dx$
$=(\frac{1+\sqrt{5}}{2}-1)+2(2-\frac{1+\sqrt{5}}{2})=\frac{5-\sqrt{5}}{2}$