Question

If $y=\{x{\}}^{\left[x\right]},$ then $\underset{0}{\overset{3}{\int }}ydx$ is equal to,where {.} and [.] are fractional part function and greatest integer function respectively.

• A. $\frac{1}{6}$
• B. $\frac{5}{6}$
• C. $\frac{7}{6}$
• D. $\frac{11}{6}$

Answer 1

The correct option is D $\frac{11}{6}$

$I=\underset{0}{\overset{3}{\int }}(x-[x]{)}^{\left[x\right]}dx$
$=\underset{0}{\overset{1}{\int }}{x}^{0}dx+\underset{1}{\overset{2}{\int }}(x-1{)}^{1}dx+\underset{2}{\overset{3}{\int }}(x-2{)}^{2}dx$
$={\left[x\right]}_0^1+{\left[\frac{(x-1{)}^2}{2}\right]}_1^2+{\frac{(x-2{)}^3}{3}|}_2^3$
$=1+\frac{1}{2}+\frac{1}{3}=\frac{11}{6}$