If [x] denotes the greatest integer function, | vedclass

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

Question

(D) We split the integral based on the definition of the greatest integer function $[x]$ in the interval $[0, 5]$:$\int_0^5 x^2[x] d x = \int_0^1 x^2(

Vedclass · Accepted Answer

$\frac{400}{3}$

Vedclass · Answer

(D) We split the integral based on the definition of the greatest integer function $[x]$ in the interval $[0, 5]$:$\int_0^5 x^2[x] d x = \int_0^1 x^2(0) d x + \int_1^2 x^2(1) d x + \int_2^3 x^2(2) d x + \int_3^4 x^2(3) d x + \int_4^5 x^2(4) d x$$= 0 + \left[\frac{x^3}{3}\right]_1^2 + 2\left[\frac{x^3}{3}\right]_2^3 + 3\left[\frac{x^3}{3}\right]_3^4 + 4\left[\frac{x^3}{3}\right]_4^5$$= \frac{1}{3}(8 - 1) + \frac{2}{3}(27 - 8) + \frac{3}{3}(64 - 27) + \frac{4}{3}(125 - 64)$$= \frac{7}{3} + \frac{38}{3} + 37 + \frac{244}{3}$$= \frac{7 + 38 + 111 + 244}{3} = \frac{400}{3}$

If $[x]$ denotes the greatest integer function,then $\int_0^5 x^2[x] d x=$

Similar Questions

If $\int_{1}^{k}(3x^{2}+2x+1)dx=11$,then $k=$

The value of $\int_{0}^{20\pi} (\sin^4 x + \cos^4 x) dx$ is equal to:

If $I=\int_1^3 \sqrt{3+x+x^2} dx$,then $I$ lies in the interval

The value of $\int_{-1}^{2} \frac{|x|}{x} d x$ is

If $p(x)$ is a cubic polynomial with $p(1)=3, p(0)=2$ and $p(-1)=4$,then $\int_{-1}^1 p(x) dx$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module