Let $\phi : [0,1] \longrightarrow \mathbb{R}$ be a continuous function such that $$\int_0^1 \phi(t) e^{-at}\,\mathrm{d}t = 0$$ for every $a \in \mathbb{R}_+$. Show that for every non-negative integer $n$, $$\int_0^1 \phi(t) t^n\,\mathrm{d}t = 0$$
Let $\phi : [0,1] \longrightarrow \mathbb{R}$ be a continuous function such that
$$\int_0^1 \phi(t) e^{-at}\,\mathrm{d}t = 0$$
for every $a \in \mathbb{R}_+$. Show that for every non-negative integer $n$,
$$\int_0^1 \phi(t) t^n\,\mathrm{d}t = 0$$