Let [·] denote the greatest integer function. Then ∫_ -(π)/(2) ^ (π)/(2) ((12(3+[x]))/(3+[ x]+[ x])) d x is equal to :

Correct answer: C. 11 π+2 I = ∫_ -π/2 ^ π/2 (12(3 + [x]) \, dx)/(3 + [ x] + [ x]) I = ∫_ -π/2 ^ -1 (12(1) \, dx)/(2) + ∫_ -1 ^ 0 (12(2) \, dx)/(2) + ∫_ 0 ^ 1 (12(3) \, dx)/(3) + ∫_ 1 ^ π/2 (12(4) \, dx)/(3) I = 6((π)/(2) - 1) + 12(0 + 1) + 12(1 - 0) + 16((π)/(2) - 1) I = 3π - 6 + 12 + 12 + 8π - 16 I = 11π + 2

JEE Main 202628 JanuaryEvening Shift

Exam: JEE Main
Year: 2026
Date: 28 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Definite Integration

JEE Main 2026 · 28 January · Evening Shift · Mathematics · Definite Integration

Let [·] denote the greatest integer function. Then ∫_ -(π)/(2) ^...

Let

[\cdot]

denote the greatest integer function. Then

\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\right) \mathrm{d} x

is equal to :

Options

A $13 \pi+1$
B $12 \pi+5$
C $11 \pi+2$
D $15 \pi+4$

Correct answer

C. $11 \pi+2$

Step-by-step solution

I = \int_{-\pi/2}^{\pi/2} \dfrac{12(3 + [x]) \, dx}{3 + [\sin x] + [\cos x]}

I = \int_{-\pi/2}^{-1} \dfrac{12(1) \, dx}{2} + \int_{-1}^{0} \dfrac{12(2) \, dx}{2} + \int_{0}^{1} \dfrac{12(3) \, dx}{3} + \int_{1}^{\pi/2} \dfrac{12(4) \, dx}{3}

I = 6\left(\dfrac{\pi}{2} - 1\right) + 12(0 + 1) + 12(1 - 0) + 16\left(\dfrac{\pi}{2} - 1\right)

I = 3\pi - 6 + 12 + 12 + 8\pi - 16

I = 11\pi + 2

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