Let $f:[0,1]\times [0,1]\to\mathbb{R}$ be a continuous function. Suppose that for any $t_0\in[0,1]$, $s\mapsto f(t_0,s)$ is a smooth function of $s\in[0,1]$, and that for any $s_0\in[0,1]$, $t\mapsto f(t,s_0)$ is a smooth function of $t\in[0,1]$.
Define $F:[0,1]\to\mathbb{R}$ by $F(t)=\int_0^1f(t,s)ds$. In general, $F$ seems not to be a smooth function of $t$. What might be a good condition on $f$, as weak as possible, to make $F$ smooth in $t$?