In proving Taylor's theorem with remainder (in Tu's Manifold version) I'm stuck at showing that $\int^1_0 \frac{\partial f}{\partial x^i} (p+t(x-p))dt$ is smooth. Here $f$ is smooth and it is defined on $\mathbb{R^n}$.
Now the naive approach is to use Feynmann integration rule (or Leibnitz rule) but here we do not know if the derivative of integrand is bounded in $x$ for every $t\in[0,1]$, so we cannot bound the derivative with a nice integrable $g(t)$.
In the proof of Leibnitz rule, I don't know how to modify dominated convergence theorem. This should be an obvious question but I'm not sure how to show it. Thank you.