Definite Integration · Mathematics · IAT (IISER)

If $p(t)=\frac{t(t-1) \cdots(t-2019)}{2019!}$, then the value of

$$ \int_0^1\left(\frac{1}{t+1}+\frac{1}{t+2}+\cdots+\frac{1}{t+2020}\right) p(-t-1) d t $$

is:

$F(x)=\int_0^{e^x}\left(t^3+2 t^2-t-2\right) d t$, then for how many real numbers $x$ does $F^{\prime}(x)=0$ ?