From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$. We are also given an infinitely differentiable function $g : [0,1] \rightarrow \mathbb{R}$. We admit that there exist real numbers $d, d' \in \mathbb{R}$ such that, as $a \rightarrow +\infty$, $$\int_0^a \cos(x^2) \mathrm{d}x = \frac{\sqrt{2\pi}}{4} + \frac{d}{a} \sin(a^2) + \frac{d'}{a^3} \cos(a^2) + O\left(\frac{1}{a^5}\right)$$ Show that, as $t \rightarrow +\infty$, $$\int_{x_0}^1 g(x) \sin(tf(x)) \mathrm{d}x = g(x_0) \int_{x_0}^1 \sin(tf(x)) \mathrm{d}x + O\left(\frac{1}{t}\right)$$
From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$. We are also given an infinitely differentiable function $g : [0,1] \rightarrow \mathbb{R}$.
We admit that there exist real numbers $d, d' \in \mathbb{R}$ such that, as $a \rightarrow +\infty$,
$$\int_0^a \cos(x^2) \mathrm{d}x = \frac{\sqrt{2\pi}}{4} + \frac{d}{a} \sin(a^2) + \frac{d'}{a^3} \cos(a^2) + O\left(\frac{1}{a^5}\right)$$
Show that, as $t \rightarrow +\infty$,
$$\int_{x_0}^1 g(x) \sin(tf(x)) \mathrm{d}x = g(x_0) \int_{x_0}^1 \sin(tf(x)) \mathrm{d}x + O\left(\frac{1}{t}\right)$$