Question

The value of ${\int }_0^{2}f\left(x\right)dx$, where $\{\begin{array}{cc}0,& \text{when}x=\frac{n}{n+1},n=1,2,3,\dots \\ 1,& elsewhere\end{array}$ is equal to

• A. 1
• B. 2
• C. 3
• D. none of these

Answer 1

The correct option is B 2

${\int }_0^{2}f\left(x\right)dx$
$={\int }_0^{\frac{1}{2}}f\left(x\right)dx+{\int }_{\frac{1}{2}}^{1}f\left(x\right)dx+{\int }_1^{\frac{2}{3}}f\left(x\right)dx+{\int }_{\frac{2}{3}}^{\frac{3}{4}}f\left(x\right)dx+{\int }_{\frac{3}{4}}^{\frac{4}{5}}f\left(x\right)dx$
$=2$