Use the method of Fourier analysis to calculate the following integral:
$$ \int_{0}^{\infty} \frac{\cos x}{1+4x^2} \operatorname{d} x .$$
Could someone help about this question? what skills should I use? Should I change the $\cos$ function to $\exp$?
One can see $\int\limits_0^{\infty}\frac{\cos(x)}{1+4x^2}dx=\frac{1}{2}\int\limits_{-\infty}^{\infty}\frac{\cos(x)}{1+4x^2}dx$ as the Fourier trasform of $f(x)=\frac{\sqrt{2\pi}}{2(1+4x^2)}$ valuated in 1, which gives $\frac{\pi}{4\sqrt{e}}$.