Evaluate the Integral $$\int_{0}^{\infty} x^{s-1} \cos(2\pi ax) \mathrm{d}x$$

How to evaluate

\int\limits_{0}^{\infty} x^{s-1} \cos(2\pi ax) \mathrm{d}x
\tag{1}
\label{eq:mtc-1}

was part of a question posed at Mathematics Stack Exchange. Note that this is the Mellin transform of the indicated cosine function.

The original answer that I provided required some rather questionable steps regarding the limits of integration, so here I provide another solution that avoids such difficulties.

In Volume 2 of Higher Transcendental Functions (Bateman Manuscript), Section 9.10, Equation 1 we have a generalization of the fresnel integrals attributed to Bohmer:
\begin{align}
\mathrm{C}(x,a) &= \int\limits_{x}^{\infty} z^{a-1} \cos(z) \mathrm{d}z \\
&= \frac{1}{2} \Big[\mathrm{e}^{i\pi a/2} \Gamma(a,-ix) + \mathrm{e}^{-i\pi a/2} \Gamma(a,ix)\Big]
\end{align}

Thus

\mathrm{C}(0,a) = \int\limits_{0}^{\infty} z^{a-1} \cos(z) \mathrm{d}z
= \Gamma(a) \cos\left(\frac{\pi}{2}a\right)

For our integral, let $$z=2\pi ax$$:

\int\limits_{0}^{\infty} x^{s-1} \cos(2\pi ax) \mathrm{d}x
= (2\pi a)^{-s} \int\limits_{0}^{\infty} z^{s-1} \cos(z) \mathrm{d}z
= (2\pi a)^{-s} \Gamma(s) \cos\left(\frac{\pi}{2}s\right)