Something Must Be Wrong at this Floor

From Property

$\int _{ a }^{ b }{ f\left( x \right) d\left( g\left( x \right) \right) } +\int _{ a }^{ b }{ g\left( x \right) d\left( f\left( x \right) \right) } =f\left( b \right) g(b)-f\left( a \right) g\left( a \right)$

Now Put

$f\left( x \right) ={ x }^{ 2 }+1\quad and\quad g\left( x \right) =\left\lfloor x \right\rfloor \quad a=0\quad ,\quad b=2$

$\int _{ 0 }^{ 2 }{ \left( { x }^{ 2 }+1 \right) d\left( \left\lfloor x \right\rfloor \right) } +\int _{ 0 }^{ 2 }{ \left\lfloor x \right\rfloor d\left( { x }^{ 2 }+1 \right) } =\left( { 2 }^{ 2 }+1 \right) \left\lfloor 2 \right\rfloor -\left( { 0 }^{ 2 }+1 \right) \left\lfloor 0 \right\rfloor$

$\int _{ 0 }^{ 2 }{ \left( { x }^{ 2 }+1 \right) d\left( \left\lfloor x \right\rfloor \right) } +=10-0-\int _{ 0 }^{ 2 }{ \left\lfloor x \right\rfloor d\left( { x }^{ 2 }+1 \right) }$

$=10-\int _{ 0 }^{ 2 }{ \left\lfloor x \right\rfloor d\left( { x }^{ 2 }+1 \right) } \\ =10-\int _{ 0 }^{ 2 }{ 2x\left\lfloor x \right\rfloor d\left( x \right) }$

$=10-2\left( \int _{ 0 }^{ 1 }{ 0.xd\left( x \right) } +\int _{ 1 }^{ 2 }{ xd\left( x \right) } \right) \\ =10-2\left( \int _{ 1 }^{ 2 }{ xd\left( x \right) } \right)$

$=10-2\left( \int _{ 0 }^{ 1 }{ 0.xd\left( x \right) } +\int _{ 1 }^{ 2 }{ xd\left( x \right) } \right) \\ =10-2\left( \int _{ 1 }^{ 2 }{ xd\left( x \right) } \right) \\ =10-2\left( \frac { { 2 }^{ 2 }-{ 1 }^{ 2 } }{ 2 } \right) \\ =10-3=7$

We have:

$\begin{aligned} \int_{0}^{2} (x^2+1)d\left\lfloor x\right\rfloor&=\lim_{n\to +\infty}\sum_{k=1}^{2n} (x_k^2+1)\left(\left\lfloor \dfrac{k}{n}\right\rfloor-\left\lfloor \dfrac{k-1}{n}\right\rfloor\right)&;x_k\in\left(\dfrac{k-1}{n};\dfrac{k}{n}\right)\\ &=\lim_{n\to +\infty}\left((x_n^2+1)+(x_{2n}^2+1)\right)\\ &=2+5=\boxed{7} \end{aligned}$