Question

$\int_{0}^{1} x^{2}[x]dx$

Answer 1

0

Answer 2

For $x$ in $[0,1)$, the greatest integer function $[x] = 0$. Therefore, the integrand becomes $x^2 \cdot 0 = 0$. The point $x = 1$ does not contribute to the integral (since it is just a single point). Hence,

$$\int_{0}^{1} x^2 [x]\,dx = \int_{0}^{1} 0\,dx = 0.$$