For 3a we are asked to take the Fourier transform of

$$ (x^2 + a^2) ^ {-1} $$ For $ a> 0 $

i.e. to find $ \tilde{f(k)} = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-ikx} f(x) dx = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-ikx} (x^2 + a^2) ^ {-1} dx $ and using Euler's identity, we can decompose the integral into cosines and sines (unlike in problems 2, where the trick seemed to lie doing the reverse) giving us:

$$ \tilde{f(k)} = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) (coskx - isinkx) \frac{1}{x^2 + a^2} dx $$

The denominator is even, the first term is even , so we'll keep it (as it is even), the second term is odd, with an even denominator. Integrating over all space means that the integral of any odd function will be 0, leaving only the even function integral. Furthermore, since even functions are symmetric about x = 0, we can rewrite the integral:

$$ \tilde{f(k)} = \frac{2}{\sqrt{2\pi}} \int_{0}^{\infty}\frac{coskx}{x^2 + a^2} dx $$

From here, I'm not quite sure how to approach this integral, as I'd normally try a trig substitution, but it seems like it would only complicate this...

Am I missing something obvious? What steps lack justification/ are incorrect?

NOTE: This question also applies to the approach in general for these problems in 3, as 3b,3c, and 3d are complications on this/add on more steps with integration by parts, so I'd also like to know if this even is the right treatment for this class of problems