Question

If $y=[x{]}^2+2\{x\left\}\right[x]+\frac{\sum _{r=1}^{100}\{x+r{\}}^2}{100}$, then $\int y\cdot dx=$
(where [.] and {.} are greatest integer function and fractional part function respectively and $c$ is integration constant)

• A. $\frac{x^2}{2}+c$
• B. $x^3+c$
• C. $\frac{x^3}{3}+c$
• D. $x+c$

Answer 1

The correct option is C $\frac{x^3}{3}+c$

Given:
$y=[x{]}^2+2\{x\left\}\right[x]+\frac{\sum _{r=1}^{100}\{x+r{\}}^2}{100}$
$\Rightarrow y=[x{]}^2+2\{x\left\}\right[x]+\frac{\{x+1{\}}^2+\{x+2{\}}^2+\{x+3{\}}^2+\cdots \cdots +\{x+100{\}}^2}{100}$
$\Rightarrow y=[x{]}^2+2\{x\left\}\right[x]+\frac{\{x{\}}^2+\{x{\}}^2+\{x{\}}^2+\cdots \cdots +\{x{\}}^2}{100}$
$\Rightarrow y=[x{]}^2+2\{x\left\}\right[x]+\frac{100\{x{\}}^2}{100}$
$\Rightarrow y=[x{]}^2+2\{x\left\}\right[x]+\{x{\}}^2$
$\Rightarrow y=\left(\right[x]+\{x\}{)}^2=x^2$
$\therefore \int ydx=x^{2}dx=\frac{x^3}{3}+c$