Question:
The value of \( \int_0^\infty \left[ \sqrt{x} \right] dx \), where \( \left[ . \right] \) is the greatest integer function, is?
The value of \( \int_0^\infty \left[ \sqrt{x} \right] dx \), where \( \left[ . \right] \) is the greatest integer function, is?
Show Hint
When dealing with the greatest integer function, split the integral into intervals where the function is constant and integrate piecewise.
Updated On: Jan 12, 2026
- \( 2 - \sqrt{2} \)
- \( \sqrt{2} - 2 \)
- \( 2 \)
- \( 2 \sqrt{2} \)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
The greatest integer function gives integer values, and the integral evaluates to 2 after solving by breaking it into intervals where the greatest integer remains constant.
Was this answer helpful?
0
0
Top VITEEE Mathematics Questions
- The product of all values of $(\cos \alpha + i \sin \alpha)^{3/5}$ is equal to
- A football is inflated by pumping air in it. When it acquires spherical shape its radius increases at the rate of 0.02 cm/s. The rate of increase of its volume when the radius is 10 cm is ___________$\pi$ cm/s
- The interval in which the function $f\left(x\right) = \frac{4x^{2}+1}{x}$ is decreasing is :
- If the normal to the curve $y = f(x)$ at $(3,4)$ makes an angle $\frac{3 \pi}{4}$ with the positive x-axis, then $f'(3)$ is equal to
- The principal value of $sin^{-1} \left(sin \frac{5\pi}{3}\right)$ is
View More Questions
Top VITEEE Some Properties of Definite Integrals Questions
- The value of the integral $\int\limits ^{1/2}_{-1/2}\left[\left(\frac{x+1}{x-1}\right)^{^2}+\left(\frac{x+1}{x-1}\right)^{^2}-2\right]^{^{1/2}}\:\:dx$ is
- The value of the integral $\int\limits_{0}^{\frac{\pi}{2}} log \left(tan\,x\right)dx=$
- The area lying in the first quadrant and bounded by the circle \( x^2 + y^2 = 4 \), the line \( x = \sqrt{3}y \), and the \( x \)-axis is:
- The area bounded by \( f(x) = x^2, 0 \leq x \leq 1 \), and \( g(x) = x + 2, 1 \leq x \leq 2 \) and x-axis is
- What is the area of a loop of the curve \( y = \sin 30^\circ \)?
View More Questions
Top VITEEE Questions
- A motor of power $P_0$ is used to deliver water at a certain rate through a given horizontal pipe. To increase the rate of flow of water through the same pipe $n$ times, the power of the motor is increased to $P_1$. The ratio of $P_1 $ to $P_0$ is
- A $30\,V-90\,W$ lamp is operated on a $120\,V$ DC line. A resistor is connected in series with the lamp in order to glow it properly. The value of resistance is
- Which of the following is not a characteristic of transition elements ?
- Most crystals show good cleavage because their atoms, ions or molecules are
- Which of the following sets consists only of essential amino acids ?
View More Questions