Definite Integration
Integral of Floor of Trig Functions
nta_pyq_2023_apr
Grade 12
Question:
Let $[t]$ denote the greatest integer $\leq t$. Then $\dfrac{2}{\pi}\displaystyle\int_{\pi/6}^{5\pi/6}(8[\csc x]-5[\cot x])\,dx$ is equal to _______
Step-by-Step Solution
Key Concept: On $(\frac{\pi}{6},\frac{5\pi}{6})$: $[\csc x]=1$ since $\csc x\in[1,2)$. For $[\cot x]$: split at $\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4}$ and evaluate each piece.
$8I_1-5I_2=8\cdot\frac{2\pi}{3}-5\cdot(-\frac{\pi}{3})=\frac{7\pi}{1}$... $\frac{2}{\pi}\cdot 7\pi=14$.
Correct Answer: 14