Derivative of integral with floor function

Let $F:[0, 2]\to\mathbb{R}$ be the function defined by $$F(x)=\int_{x^2}^{x+2} e^{x[t]}\,dt,$$ where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the derivative of $F$ at $x=1$ equals