Question

The value of integral $\frac{\underset{0}{\overset{10}{\int }}\left[x\right]dx}{\underset{0}{\overset{10}{\int }}\left\{x\right\}dx}$
( [.] and {.} represents the greatest integer function and the fractional part funtion respectively ) is

• A. $10$
• B. $9$
• C. $11$
• D. $8$

Answer 1

The correct option is B $9$

$\underset{0}{\overset{n}{\int }}\left[x\right]dx=\underset{0}{\overset{1}{\int }}0dx+\underset{1}{\overset{2}{\int }}1dx+\underset{2}{\overset{3}{\int }}2dx+\cdots +\underset{9}{\overset{10}{\int }}9dx\Rightarrow \underset{0}{\overset{n}{\int }}\left[x\right]dx=1+2+3+\cdots +9\Rightarrow \underset{0}{\overset{n}{\int }}\left[x\right]dx=\frac{9\times 10}{2}=45$

$\underset{0}{\overset{10}{\int }}\left\{x\right\}dx=10\times \underset{0}{\overset{1}{\int }}\left\{x\right\}dx$
$\Rightarrow \underset{0}{\overset{10}{\int }}\left\{x\right\}dx=10\times \frac{1}{2}=5$

$\frac{\underset{0}{\overset{10}{\int }}\left[x\right]dx}{\underset{0}{\overset{10}{\int }}\left\{x\right\}dx}=\frac{45}{5}=9$