Definite Integration — NDA Mathematics PYQs

Consider the following for the two (02) items that follow :
Let $f(x)=[x^2]$, where $[\cdot ]$ is the greatest integer function.

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

Consider the following for the two (02) items that follow :
Let $f(x)=[x^2]$, where $[\cdot ]$ is the greatest integer function.

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

What is ${\int }_n^{n+1}(x-[x])dx$, where $[\cdot ]$ is the greatest integer function and $n$ is a 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?