Question

Let $\left[x\right]$ and $\left\{x\right\}$ be the integer part and fractional part of a real number $x$ respectively. The value of the integral ${\int }_0^5\left[x\right]\left\{x\right\}dx$ is

• A. $5/2$
• B. $5$
• C. $34.5$
• D. $35.5$

Answer 1

The correct option is A $5/2$

${\int }_0^5\left[x\right]\left\{x\right\}dx$

$={\int }_0^{1}0dx+{\int }_1^2(x-1)dx+{\int }_2^{3}2(x-2)dx+{\int }_3^{4}3(x-3)dx+{\int }_4^{5}4(x-4)dx$

$=0+{[\frac{x^2}{2}-x]}_1^2+{[x^2-4x]}_2^3+{[\frac{3x^2}{2}-9x]}_3^4+[2x^2-16x{]}_4^5$

$=2-2-\frac{1}{2}+1+9-12-4+8+24-36-13.5+27+50-80-32+64$

$=0.5+9-16+32-49.5+27-30+32$

$=41+59-49-46$

$=5$