Question

Let $\left[x\right]$ denote the greatest integer less than or equal to $x$, then the value of the integral ${\int }_{-1}^1(|x|-2[x\left]\right)dx$ is equal to

• A. $3$
• B. $2$
• C. $-2$
• D. $-3$

Answer 1

Answer: (A)

$3$

Let $I={\int }_{-1}^1(|x|-2[x\left]\right)dx$
$={\int }_{-1}^{=}(|z|-2[x\left]\right)dx{\int }_0^1(|x|-2[x\left]\right)dx$
$={\int }_{-1}^0(-x-2(-1\left)\right)dx+{\int }_0^1(x-2(0\left)\right)dx$
$={\int }_{-1}^0(-x+2)dx+{\int }_0^{1}xdx$
$={(-\frac{x^2}{2}+2x)}_{-1}^0+{\left[\frac{x^2}{2}\right]}_0^1$
$=-(-\frac{1}{2}+2(-1\left)\right)+\frac{1}{2}$
$=\frac{1}{2}+2+\frac{1}{2}=1+2=3$