During my research I ran into the following type of an oscillatory integral, for some values of nonzero reals $a,b$:
$f(R):=\int_{0}^{R} e^{2 \pi i (ar^2 + br)} dr$
and I am interested in finding a good estimate for it when $R>0$ is large, in terms of $R$ and the two parameters $a,b$. I expect $\frac{1}{R}f(R)$ to converge as $R \to \infty$ uniformly in $b$, but a more quantitative estimate will be helpful. I guess this is classical but since I know very little in this area I'm not sure where to find such results.
EDIT: Thanks for your help. In the end I just used a version of van der Corput's lemma, but the answers offered here were still interesting to read.