Do you really need "contour" here? (2)

Calculus Level 4

$\large \int_{-\infty}^{\infty} \dfrac{\cos x}{1+x^{2}} \, dx$

Find the value of the closed form of the above integral.
Give your answer to 3 decimal places.

Bonus:

Solve this question without using contour integration.
Generalise $\displaystyle \int_{-\infty}^{\infty} \dfrac{\cos(x)}{a^{2}+x^{2}} dx$ , where $a$ is a constant.