Question

The value of $\underset{0}{\overset{100}{\int }}(x-[x\left]\right)dx$,
where $[\cdot ]$ represents the greatest integer function, is

• A. $10$
• B. $25$
• C. $50$
• D. $100$

Answer 1

The correct option is C $50$

Let $I=\underset{0}{\overset{100}{\int }}(x-[x\left]\right)dx$
$\Rightarrow I=\underset{0}{\overset{100}{\int }}\left\{x\right\}dx$
Using the properties:
$\underset{0}{\overset{nT}{\int }}f\left(x\right)dx=n\underset{0}{\overset{T}{\int }}f\left(x\right)dx\left[T\text{is the period of}f\right(x\left)\right]$
Here, the period of $\left\{x\right\}$ is $1$, so
$I=100\underset{0}{\overset{1}{\int }}\left\{x\right\}dx\Rightarrow I=100\underset{0}{\overset{1}{\int }}xdx\therefore I=100\times \frac{1}{2}=50$