In the books Analytic combinatorics, the authors write (p. 757, eq (36)), with $k_n=n^{1/10}$, that
\begin{align*} I^{(1)}_{n}:&=\int_{-k_{n}/\sqrt{n}}^{k_{n}/\sqrt{n}}\cos^n(x)\ dx\\ &=\frac{1}{\sqrt{n}}\int_{-k_{n}}^{k_{n}}e^{-w^2/2}\exp\left(\mathcal{O}(n^{-1}w^{4})\right)\ dw\\ &=\frac{1}{\sqrt{n}}\int_{-k_{n}}^{k_{n}}e^{-w^2/2}\left(1+\mathcal{O}(n^{-1}w^{4})\right)\ dw\\ &=\frac{1}{\sqrt{n}}\int_{-k_{n}}^{k_{n}}e^{-w^2/2}\ dw+\mathcal{O}(n^{-3/5}).\\ \end{align*} I have a problem with the last equality where it seems to me that the last estimate should be $\mathcal{O}(n^{-3/2})$ instead of $\mathcal{O}(n^{-3/5})$ as $w^4 e^{-w^2}$ is integrable over $\mathbb{R}$. It seems that I miss something, but I could not find out what it is...
Thanks