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

Calculus Level 4

cosx1+x2dx\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 cos(x)a2+x2dx\displaystyle \int_{-\infty}^{\infty} \dfrac{\cos(x)}{a^{2}+x^{2}} dx, where aa is a constant.

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