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

Mathematics · Definite Integration

JEE Main 2026 — 22 January, Morning Shift — Question 14

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[x]+4}\right) d x$ , where $[\bullet]$ denotes the greatest integer function, is

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

Answer: C

Step-by-step solution

$I=\int_{-\pi / 2}^{\pi / 2} \frac{1}{[x]+4} d x$

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

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

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

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Exam: JEE Main 2026
Paper: 22 January, Morning Shift
Subject: Mathematics
Chapter: Definite Integration
Topic: Evaluation of Definite Integrals

Let f:[1, infty) arrow mathbbR be a differentiable function, If 6 int 1^x f(t) d t=3 x f(x)+x^3-4 for all x geq 1 , then the value of…

Answer: C

Step-by-step solution

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