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

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Aicharya · Accepted Answer

$\begin{aligned} \int_0^5 x^2[x] \mathrm{d} x= & \int_0^1 x^2[x] \mathrm{d} x+\int_1^2 x^2[x] \mathrm{d} x \ & +\int_2^3 x^2[x] \mathrm{d} x+\int_3^4 x^2[x]+\int_4^5 x^2[x] \mathrm{d} x\end{aligned}$ $\begin{aligned}=\int_0^1 x^2(0) \mathrm{d} x+\int_1^2 x^2(1) \mathrm{d} x & +\int_2^3 x^2(2) \mathrm{d} x \ & +\int_3^4 x^2(3) \mathrm{d} x+\int_4^5 x^2(4) \mathrm{d} x\end{aligned}$ $\begin{aligned} & =\left[\frac{x^3}{3}ight]_i^2+2\left[\frac{x^3}{3}ight]_2^3+3\left[\frac{x^3}{3}ight]_3^4+4\left[\frac{x^3}{3}ight]_4^5 \ & =\frac{1}{3}(8-1)+\frac{2}{3}(27-8)+(64-27)+\frac{4}{3}(125-64) \ & =\frac{400}{3}\end{aligned}$

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

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