What is $\int _{n}^{n+1}(x-[x])dx$, where [.] is the greatest integer function and n is natural number?

If $I_{1}=\int _{e}^{e^{2}}\frac{dx}{ln~x}$ and  $I_{2}=\int _{1}^{2}\frac{e^{x}}{x}dx$  then which one of the following is correct?

Consider the following for the two (02) items that follow:
Let f(x) = [x²] where [.] is the greatest integer function.

What  $\int_{\sqrt{2}}^{\sqrt{3}} f(x) dx$ equal to?

Consider the following for the two (02) items that follow:
Let f(x) = [x²] where [.] is the greatest integer function.

$\int_{\sqrt{2}}^{2} f(x) dx$  is equal to ?