Question

Let $f\left(x\right)=x-\left[x\right]$, for every real number $x$ (where $\left[x\right]$) is integral part of $x$). Then the value of ${\int }_{-1}^{1}f\left(x\right)dx$, is equal to

• A. 0
• B. 1
• C. 2
• D. none of these

Answer 1

Answer: (B)

1

Here, $f\left(x\right)=x-\left[x\right]$
${\int }_{-1}^{1}f\left(x\right)dx={\int }_{-1}^{1}x-\left[x\right]dx$
$={\int }_{-1}^{1}xdx-{\int }_{-1}^1\left[x\right]dx$
$={\left[\frac{x^2}{2}\right]}_{-1}^1-[{\int }_{-1}^0\left[x\right]dx+{\int }_0^1\left[x\right]dx]$
$=0-[{\int }_{-1}^0(-1)dx+{\int }_0^{1}0dx]$
$\Rightarrow I=1$