Question

${\int }_0^8\left[x^{\frac{1}{3}}\right]dx$ is equal to (where $[.]$ denotes the greatest integers function)

• A. $8$
• B. $7$
• C. $6$
• D. $5$

Answer 1

The correct option is A $8$

${\int }_0^8\left[x^{\frac{1}{3}}\right]dx$

$0<x<8$

$=0<x^{1/3}<2$

$=0<\left[x^{\frac{1}{3}}\right]<2$

$={\int }_0^1\left[x^{\frac{1}{3}}\right]dx+{\int }_1^8\left[x^{\frac{1}{3}}\right]dx$

$={\int }_0^{1}0dx+{\int }_1^{8}1.dx$

$=0+{\left[x\right]}_1^8$

$=8-1$

$=7$