<s>[INST] How fast does $a_n:=\int_{1/\sqrt{2}}^1 \frac{dx}{\left(\frac{1}{2}+x^2\right)^{n+1/2}}$ decay as $n \to \infty$? Let
\begin{equation}
a_n:=\int_{1/\sqrt{2}}^1 \frac{dx}{\left(\frac{1}{2}+x^2\right)^{n+1/2}}
\end{equation}
for $n \in \mathbb{N}$.
Then, $a_n$ is clearly a monotone decreasing sequence of positive numbers and by the Dominated Convergence Theorem, $a_n \to 0^+$ as $n \to \infty$.
However, I have some difficulty estimating how fast $a_n$ decays. For example, $\frac{a_n}{f(n)}=O(1)$ as $n \to \infty$ for some polynomial $f$?
Could anyone please provide insight into the decay rate of $a_n$?
 [/INST] Alternative solution:
With the substitution $x = \sqrt y$, we have
$$a_n = \int_{1/2}^1 \frac{1}{(1/2 + y)^{n+1/2}}\frac{1}{2\sqrt y}\,\mathrm{d}y.$$
It is easy to prove that, for all $y \in [1/2, 1]$,
$$\frac{\sqrt 2}{2} - \frac{\sqrt 2}{2}(y - 1/2) \le \frac{1}{2\sqrt y} \le \frac{\sqrt 2}{2}.$$
We have
$$\int_{1/2}^1 \frac{\frac{\sqrt 2}{2}}{(1/2 + y)^{n+1/2}}\,\mathrm{d}y - \int_{1/2}^1 \frac{\frac{\sqrt 2}{2}(y - 1/2)}{(1/2 + y)^{n+1/2}}\,\mathrm{d}y \le a_n \le \int_{1/2}^1 \frac{\frac{\sqrt 2}{2}}{(1/2 + y)^{n+1/2}}\,\mathrm{d}y$$
which results in
\begin{align*}
 &\frac{\sqrt 2}{2n-1} - \frac{\sqrt 2}{2n-1}(2/3)^{n-1/2} - \frac{2\sqrt 2}{4n^2 - 8n + 3} + \frac{(n + 3/2)\sqrt 2}{4n^2 - 8n + 3}(2/3)^{n-1/2}\\ 
 \le{}& a_n \le \frac{\sqrt 2}{2n-1} - \frac{\sqrt 2}{2n-1}(2/3)^{n-1/2}.
\end{align*}
Thus,
$$a_n = \frac{\sqrt 2}{2n} + O\left(\frac{1}{n^2}\right).$$
</s>