Question

The value of ${\int }_0^4\left\{x\right\}dx$, (where $\left\{x\right\}$ denotes fractional part of $x$) is equal to

• A. $\frac{4}{3}$
• B. $\frac{5}{3}$
• C. $\frac{7}{3}$
• D. None of these

Answer 1

The correct option is D None of these

$I={\int }_0^4\left\{x\right\}dx$
$={\int }_0^4(x-[x\left]\right)dx$
$={\int }_0^{4}xdx-{\int }_0^4\left[x\right]dx$
$=(\frac{x^2}{2}{)}_0^4-({\int }_0^{1}0dx+{\int }_1^{2}1dx+{\int }_2^{3}2dx+{\int }_3^{4}3dx)$
$=8-\left(\right[x{]}_1^2+[2x{]}_2^3+[3x{]}_3^4)$
$\Rightarrow I=2$