Let $[t]$ denote the greatest integer function. If $\displaystyle\int_0^{2.4}[x^2]\,dx=\alpha+\beta\sqrt{2}+\gamma\sqrt{3}+\delta\sqrt{5}$, then $\alpha+\beta+\gamma+\delta$ is equal to
Step-by-Step Solution
Key Concept: Split: $[x^2]=0$ on $[0,1)$, $1$ on $[1,\sqrt{2})$, $2$ on $[\sqrt{2},\sqrt{3})$, $3$ on $[\sqrt{3},2)$, $4$ on $[2,\sqrt{5})$, $5$ on $[\sqrt{5},2.4)$. Integrate each piece.
$\int=9-\sqrt{2}-\sqrt{3}-\sqrt{5}$. $\alpha+\beta+\gamma+\delta=6$.
Correct Answer: 6