Definite Integration
GIF in Denominator — Piecewise Integration
nta_pyq_2026_jan
Grade 12
Question:
The value of $\displaystyle\int_{-\pi/2}^{\pi/2}\frac{1}{[|x|]+4}\,dx$, where $[\cdot]$ denotes the greatest integer function, is
$\dfrac{1}{60}(21\pi-1)$
$\dfrac{7}{60}(\pi-3)$
$\dfrac{7}{60}(3\pi-1)$
$\dfrac{1}{60}(\pi-7)$
Step-by-Step Solution
Key Concept: Split: $\int_{-\pi/2}^{-1}\frac{dx}{2}+\int_{-1}^{0}\frac{dx}{3}+\int_{0}^{1}\frac{dx}{4}+\int_{1}^{\pi/2}\frac{dx}{5}$. Note $[|x|]=0$ for $x\in(-1,1)$... wait: $|x|\in[0,1)$ so $[|x|]=0$ and denominator $=4$; $|x|\in[1,2)$ so $[|x|]=1$ and denominator $=5$.
$I=\dfrac{7}{60}(3\pi-1)$.
Correct Answer: 3