(B) The integral is given by $I = \int_0^2 x^2 [x] \, dx$. Since $[x]$ is the greatest integer function,we split the interval $[0, 2]$ into $[0, 1)$ a

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

(B) The integral is given by $I = \int_0^2 x^2 [x] \, dx$. Since $[x]$ is the greatest integer function,we split the interval $[0, 2]$ into $[0, 1)$ and $[1, 2]$. For $0 \le x For $1 \le x Thus,$I = \int_0^1 x^2(0) \, dx + \int_1^2 x^2(1) \, dx$. $I = 0 + \int_1^2 x^2 \, dx$. $I = \left[ \frac{x^3}{3} \right]_1^2 = \frac{2^3}{3} - \frac{1^3}{3} = \frac{8}{3} - \frac{1}{3} = \frac{7}{3}$.

Class 12 Mathematics 7-2.Definite Integral Fundamental definite integration

Difficult

If $[x]$ denotes the greatest integer less than or equal to $x,$ then the value of the integral $\int_0^2 {{x^2}[x]\,dx} $ equals (in $/3$)

A
$5$
B
$7$
C
$8$
D
$4$

Explore More

Browse All

Question Bank

All Subjects

Class 12 Questions

Class 12

Mathematics Questions

Chapter

7-2.Definite Integral

Subtopic

Fundamental definite integration

If $\int_{0}^{\frac{\pi}{3}} \frac{\tan \theta}{\sqrt{2 k \sec \theta}} d \theta = 1 - \frac{1}{\sqrt{2}}$,$(k > 0)$,then the value of $k$ is

MediumJEE MAIN 2019

View Solution

If $f(x) = \begin{cases} e^{\cos x} \sin x, & \text{for } |x| \leq 2 \\ 2, & \text{otherwise} \end{cases}$,then $\int_{-2}^{3} f(x) dx$ is equal to

MediumMHT CET 2023

View Solution

The value of $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (\sin x)^{-4} \,dx$ is

DifficultMHT CET 2024

View Solution

The value of the integral $\int_{1}^{5}[|x-3|+|1-x|] dx$ is equal to

DifficultWBJEE 2012

View Solution

If $\int_a^b x^3 dx = 0$ and $\int_a^b x^2 dx = \frac{2}{3}$,then $a$ and $b$ are respectively

EasyMHT CET 2022

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

If $[x]$ denotes the greatest integer less than or equal to $x,$ then the value of the integral $\int_0^2 {{x^2}[x]\,dx} $ equals (in $/3$)

Similar Questions

If $\int_{0}^{\frac{\pi}{3}} \frac{\tan \theta}{\sqrt{2 k \sec \theta}} d \theta = 1 - \frac{1}{\sqrt{2}}$,$(k > 0)$,then the value of $k$ is

If $f(x) = \begin{cases} e^{\cos x} \sin x, & \text{for } |x| \leq 2 \\ 2, & \text{otherwise} \end{cases}$,then $\int_{-2}^{3} f(x) dx$ is equal to

The value of $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (\sin x)^{-4} \,dx$ is

The value of the integral $\int_{1}^{5}[|x-3|+|1-x|] dx$ is equal to

If $\int_a^b x^3 dx = 0$ and $\int_a^b x^2 dx = \frac{2}{3}$,then $a$ and $b$ are respectively

Vedclass Test Series

Exam Paper Generator

Online Exam Module