Question Number 160563 by qaz last updated on 02/Dec/21
![∫x{x}[x]dx=?](https://www.tinkutara.com/question/Q160563.png)
$$\int\mathrm{x}\left\{\mathrm{x}\right\}\left[\mathrm{x}\right]\mathrm{dx}=? \\ $$
Answered by mr W last updated on 02/Dec/21
![[x]=n {x}=x−n ∫x{x}[x]dx =∫x(x−n)ndx =n((x^3 /3)−((nx^2 )/2))+C =[x]x^2 ((x/3)−(([x])/2))+C](https://www.tinkutara.com/question/Q160568.png)
$$\left[{x}\right]={n} \\ $$$$\left\{{x}\right\}={x}−{n} \\ $$$$\int{x}\left\{{x}\right\}\left[{x}\right]{dx} \\ $$$$=\int{x}\left({x}−{n}\right){ndx} \\ $$$$={n}\left(\frac{{x}^{\mathrm{3}} }{\mathrm{3}}−\frac{{nx}^{\mathrm{2}} }{\mathrm{2}}\right)+{C} \\ $$$$=\left[{x}\right]{x}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{3}}−\frac{\left[{x}\right]}{\mathrm{2}}\right)+{C} \\ $$