Please consider this image below (sorry i'm not writing it manually, but i could see it's still understandable to read):
From the last two, where did the imaginary part go? I could agree if the bounds change to $[0,\,\infty)$ because of the odd property of $\cos{ut}$ i guess? (I'm not sure). But if that so, why there's still the $2$ on the numerator, since the $2$ of the odd property cancel with the denominator? Can you tell me please.
By the way $\overline{f}(a+iu)$ is the Laplace transform of $f(t)$, and the formula that you see on the first line is the Bromwich Integral a.k.a the formula for finding the Inverse Laplace Transform.