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Evaluate \[\large \int\limits\limits_0^{\infty} \dfrac{\cos x}{x^2+1}\,dx\]

Lol I was thinking series

Can it be like a power series?

This looks fun and interesting, I really wanna work on this one and find a pretty answer

it sure feels like multiple approaches are possible...

\(|\cos x|\le 1\) so replacing \(\cos x\) by \(1\) is it ?

Yeah, it's just a guess though haha

Oh wow, that e is unexpected, but \(e \approx 2\) hahaha

Can't you use Euler's equation otherwise this seems tedious xD

oops
\[\mathcal F \left\{ \frac{1}{t^2+1} \right\} = \pi \ e^{-2 \pi |f|}\]

We can use l'hopital's rule , it ll be much quicker right ?

Ohhhh... integration by parts :P