Let f x = min x - 1 , x - 2 , … , x - 10 where t denotes the greatest integer ≤ t . Then ∫…

Given, $f\left(x\right)=\min \left\{\left[x-1\right],\left[x-2\right],\dots ,\left[x-10\right]\right\}$

$\Rightarrow f\left(x\right)=\left[x\right]-10$

Now finding ${\int }_0^{10}f\left(x\right)·dx=-10-9-8-\dots ..-1$

$=-\frac{10·11}{2}=-55$

Finding${\int }_0^{10}\left(f{\left(x\right)}^2\right)dx={10}^2+9^2+8^2+\dots +1^2$

$=\frac{10·11·21}{6}=385$

And ${\int }_0^{10}\left|f\left(x\right)\right|=10+9+8+\dots .+1$

$=\frac{10·11}{2}=55$

So, ${\int }_0^{10}f\left(x\right)dx+{\int }_0^{10}{\left(f\left(x\right)\right)}^{2}dx+{\int }_0^{10}\left|f\left(x\right)\right|dx$

$=-55+385+55=385$