Here is an exercise from Dieudonné. He suggests to "perform integrations by part".
Let $f, g$ be positive $C^\infty$ functions, $F(x)=\int_1^x f(t)dt$ and assume that
- $\int_1^\infty f(t) dt = +\infty$
- $g'(t) \neq 0$ for $t \geq 1$, $\frac{f(x)}{g'(x)}$ is non-decreasing and goes to $+\infty$ with $x$.
- $\frac{d}{dx} \left(\frac{f(x)}{g'(x)} \right) = f(x)h(x)$ where $h(x)=o(1)$ and $h'(x)=o(1)$ (so that $f(x) = o(F(x) g'(x))$ and $\log F(x) = o(g(x))$).
How do I prove the following asymptotic expansion $ \int_1^x f(t) e^{i g(t)} dt \sim \frac{f(x)}{i g'(x)} e^{i g(x)}$ when $x$ goes to infinity?
I think I did all the possible integrations by parts, in vain. I'll be grateful for any indication.