Integration#3

Define $f(t)$ as the maximum value of $k$ that satisfies the following inequality where $x\in[0,2]$ :

$x^2+kx\leq e^{x-t}$

Denote $X=\displaystyle \int_2^{\frac32+\ln \frac29} f(t)\, dt$ .

Find $\left \lfloor 10^{10} X \right \rfloor$ .

Notation: $\lfloor \cdot \rfloor$ denotes the floor function .