Calculating real integral, using complex analysis

Question

Calculate an integral: $$ \int\limits_{\mathbb{R}} \frac{\cos x \, d x}{x^4+1}. $$ May somebody demonstrate how to use complex analysis there? And also how to prove that this method might be used.

Answer 1

The given integral is the real part of the integral $\int_{\mathbb{R}}\frac{e^{ix}}{x^4+1}\,dx$ that is absolutely converging. The meromorphic function $f(z)=\frac{e^{iz}}{z^4+1}$ has simple poles at the roots of $z^4+1$ and is bounded by $\frac{C}{|z|^4}$ for any large $z$ with non-negative imaginary part. By the residue theorem, the given integral equals the real part of $2\pi i$ times the sum of the residues of $f(z)$ at $z=\exp{\frac{2\pi i}{8}}$ and $z=\exp\frac{6\pi i}{8}$ (the poles in the upper half-plane). If $w$ is one of such points,

$$\text{Res}_{z=w} f(z) = \lim_{z\to w}\frac{e^{iz}(z-w)}{z^4+1} \stackrel{dH}{=} \frac{e^{iw}}{4w^3} $$ and it follows that:

$$ \int_{\mathbb{R}}\frac{\cos(x)}{x^4+1}\,dx = \color{blue}{\frac{\pi}{\sqrt{2}}\,\exp\left(-\frac{1}{\sqrt{2}}\right)\left[\sin\left(\frac{1}{\sqrt{2}}\right)+\cos\left(\frac{1}{\sqrt{2}}\right)\right]}.$$