If $\varphi$ is a smooth function on $\mathbb{R}$, then integration by parts implies that there exists a constant $C>0$ such that $$ \Big|\int_0^1 \varphi(x)\, e^{i \lambda x}\, dx\Big|<\frac{C}\lambda $$ as $\lambda\rightarrow\infty$.

$\textbf{My question}$ is, whether one can determine the rate of decay, in terms of $\lambda$, of the oscillatory integral $$ \Big|\int_0^1 \frac{1}{\sqrt{x}}\, e^{i \lambda x}\, dx\Big|? $$ Since $\frac{1}{\sqrt{x}}\in L^1((0,1))$, it follows by the Riemann-Lebesgue Lemma that $\int_0^1 \frac{1}{\sqrt{x}}\, e^{i \lambda x}\, dx$ vanishes as $\lambda\rightarrow\infty$. However, Riemann-Lebesgue Lemma doesn't say anything about the rate of decay, hence my question.

Thanks for reading!