Definite integrals

Question

$$\int_0^{1.5}[x^2]dx$$where [.] denotes the greatest integer function, is equal to :

(1) $\sqrt{2}-2$

(2) $2 –\sqrt{2}$

(3) $2 + \sqrt{2}$

(4) None of these

What I did, I broke the function into two parts..one with limits from 0 to 1.the problem is how should I deal with the other part??Please keep the explanations as simple as possible.thanks.

Answer 1

$$\int_0^{1.5} [x^2] \mathrm{d} x=\int_0^1[x^2] \mathrm{d}x+\int_1^\sqrt{2} [x^2] \mathrm{d}x+\int_{\sqrt{2}}^{1.5} [x^2] \mathrm{d}x .$$ In the first integral, $[x^2]=0$, in the second $[x^2]=1$ and in the third $[x^2]=2$.