Question

Let $y=\{x{\}}^{\left[x\right]}$ where $\left\{x\right\}$ denotes the fractional part of $x$ and $\left[x\right]$ denotes the greatest integer less than or equal to $x.$ Then $\underset{0}{\overset{3}{\int }}ydx$ is

• A. $\frac{11}{6}$
• B. $\frac{2}{3}$
• C. $1$
• D. $\frac{5}{6}$

Answer 1

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

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