The value of ∫_ -(π)/(2) ^ (π)/(2) ((1)/([x]+4)) d x , where [·] denotes the greatest integer function, is

Correct answer: C. (7)/(60)(3 π-1) I = ∫_ -π/2 ^ π/2 (1)/([x] + 4) \, dx I = ∫_ -π/2 ^ -1 (dx)/(2) + ∫_ -1 ^ 0 (dx)/(3) + ∫_ 0 ^ 1 (dx)/(4) + ∫_ 1 ^ π/2 (dx)/(5) = (1)/(2)(-1 + (π)/(2)) + (1)/(3)(1) + (1)/(4)(1) + (1)/(5)((π)/(2) - 1) = (7π)/(20) - (7)/(60) = (7)/(60)(3π - 1)

JEE Main 202622 JanuaryMorning Shift

Exam: JEE Main
Year: 2026
Date: 22 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Definite Integration

JEE Main 2026 · 22 January · Morning Shift · Mathematics · Definite Integration

The value of ∫_ -(π)/(2) ^ (π)/(2) ((1)/([x]+4)) d x , where [·]...

The value of

\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[x]+4}\right) d x

, where

[\cdot]

denotes the greatest integer function, is

Options

A $\frac{1}{60}(21 \pi-1)$
B $\frac{7}{60}(\pi-3)$
C $\frac{7}{60}(3 \pi-1)$
D $\frac{1}{60}(\pi-7)$

Correct answer

C. $\frac{7}{60}(3 \pi-1)$

Step-by-step solution

I = \int_{-\pi/2}^{\pi/2} \dfrac{1}{[x] + 4} \, dx

I = \int_{-\pi/2}^{-1} \dfrac{dx}{2} + \int_{-1}^{0} \dfrac{dx}{3} + \int_{0}^{1} \dfrac{dx}{4} + \int_{1}^{\pi/2} \dfrac{dx}{5}

= \dfrac{1}{2}\left(-1 + \dfrac{\pi}{2}\right) + \dfrac{1}{3}(1) + \dfrac{1}{4}(1) + \dfrac{1}{5}\left(\dfrac{\pi}{2} - 1\right)

= \dfrac{7\pi}{20} - \dfrac{7}{60} = \dfrac{7}{60}(3\pi - 1)

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