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

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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 $\int_0^5 x^2[x] d x=$

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