TITLE: The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$
QUESTION [2 upvotes]: Let $n\in\mathbb N$ ($n>0$), and define the $n$th integral in the sequence $I$ to be
$$I_n = \int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx.$$
Evaluating such integrals, especially for small $n$, is essentially a straightforward exercise in complex analysis (integrate on a semicircle in the UHP using the residue theorem, send the radius to $+\infty$ and show that the integral on the "curved part" goes to zero). However, rather surprisingly the values of these integrals get progressively more non-trivial as $n$ increases:
\begin{align}
\\I_1 &= \pi \approx 3.14
\\I_2 &= \frac{\pi}{\sqrt 2}\approx 2.22
\\I_3 &= \frac{2\pi}{3}\approx 2.09
\\I_4 &= \frac{\pi}{4}\csc\frac{\pi}{8}\approx 2.05
\\I_5 &= \frac{4\pi}{5(\sqrt 5 - 1)}\approx 2.033
\\I_6 &= \frac 1 6\Gamma\left(\frac 1 {12}\right)\Gamma\left(\frac{11}{12}\right)\approx 2.023
\\    &\dots
\end{align}
Is anything currently known about:
$a)$ General closed forms for $I_n$,
$b)$ The stabilization of the numerical value of $I_n$ as $n\to\infty$? Noting that these integrals are bounded below by zero and that $I_n$ is a decreasing sequence, the limit $$\lim_{n\to\infty}I_n$$
must be finite. Can we say more (either numerically or analytically)?

REPLY [1 votes]: $\quad$ All integrals of the form $~\displaystyle\int_0^\infty\dfrac{x^{k-1}}{(x^n+a^n)^m}~dx~$ can be evaluated by substituting $x=at$ and $u=\dfrac1{t^n+1}$ , then recognizing the expression of the beta function in the new integral, and lastly 
employing Euler's reflection formula for the $\Gamma$ function to simplify the result.
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