Definite Integration
Floor Function Integration
nta_pyq_2024_apr
Grade 12
Question:
Let $[t]$ denote the largest integer $\leq t$. If $\int_0^3\left([x^2]+\left[\dfrac{x^2}{2}\right]\right)dx=a+b\sqrt{2}-\sqrt{3}-\sqrt{5}+c\sqrt{6}-\sqrt{7}$, where $a,b,c\in\mathbb{Z}$, then $a+b+c$ is equal to ________.
Step-by-Step Solution
Key Concept: Break the integral at points where $[x^2]$ or $[x^2/2]$ change value: $x=1,\sqrt{2},\sqrt{3},2,\sqrt{5},\sqrt{6},\sqrt{7},\sqrt{8},3$. Evaluate each piece.
$a=31$, $b=-6$, $c=-2$. $a+b+c=23$.
Correct Answer: 23