= Solution
A sufficient <differentiation under the integral sign> theorem is the following local domination criterion. Suppose $u(t,\cdot)$ is measurable for each $t$, $u(t_0,\cdot)$ is integrable at one point of a neighborhood $J$ of $t_0$, and for almost every $x$ the function $u(\cdot,x)$ is differentiable on $J$. Assume an integrable function $g_J$ satisfies $|\partial_tu(t,x)|\leq g_J(x)$ for $t\in J$ and almost every $x$. Then $I$ is differentiable on the interior of $J$ and
$$
\boxed{I'(t)=\int_E\partial_tu(t,x)\,dx.}
$$
The <mean value theorem> bounds the difference quotients by $g_J$, and the <dominated convergence theorem> passes their pointwise limit through the integral. Applying this criterion locally throughout $(a,b)$ proves <differentiability> there. One may impose the stronger condition that every $u(t,\cdot)$ is integrable from the outset.
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