Define $\displaystyle g: V \rightarrow \mathbb{R}$ smooth where $\displaystyle V \subseteq \mathbb{R}^n$ is open. Then define a map, for a fixed $\displaystyle x \in V$

$\displaystyle h_i(x) = \int_0^1 \frac{\partial g}{\partial x_i}(tx) dt$

(we may assume tx is in V)

Can anyone explain to me why this function is smooth?

Thanks for any help