A Calculus Problem!

The Problem:

Find $I=\displaystyle\int_{0}^{2} (x^2+1)\,d\left \lfloor x\right \rfloor$

My Doubt:

There seem two probable approaches to the question but both of them yield different results. Both seem to be mathematically correct and hence the confusion.

Approach 1:

We know that,

$g(x). f'(x) + f(x). g'(x) = \large \frac{d\left(f(x). g(x)\right)}{dx}$

Integrating the above expression we get:

$\displaystyle\int_{a}^{b} g(x)\,df(x) + \displaystyle\int_{a}^{b} f(x)\,dg(x) = f(b).g(b)-f(a).g(a)$

Using the above property to solve the integration we get the answer as $\boxed{7}$

Approach 2:

We can write

$I=\displaystyle\int_{0}^{1} (x^2+1)\,d\left \lfloor x\right \rfloor +\displaystyle\int_{1}^{2}(x^2+1)\,d\left \lfloor x\right \rfloor$

But $\left \lfloor x\right \rfloor$ assumes constant values of $\left \lfloor x\right \rfloor=0$ and $\left \lfloor x\right \rfloor=1$ in the respective intervals. And hence in both cases $d\left \lfloor x\right \rfloor=0$ and therefore $I=0$

I genuinely can't understand what's the correct method. I will really be grateful if someone can explain it to me.

Thank You!