Is it necessary to use the Gamma function in integral form and then make the appropriate changes.... probably not, but is one of the shortest methods to choose.
Consider the integral
\begin{align}
I_{n} = \int_{0}^{\infty} \cos(x^{2n}) \, dx
\end{align}
and make the change of variables $t = x^{2n}$ which leads to
\begin{align}
I_{n} = \frac{1}{2n} \, \int_{0}^{\infty} \cos(t) \, t^{\frac{1}{2n} - 1} \, dt.
\end{align}
Now using the integral
\begin{align}
\int_{0}^{\infty} \cos(at) \, t^{p-1} \, dt = \frac{\Gamma(p)}{a^{p}} \, \cos\left( \frac{p \pi}{2} \right)
\end{align}
then $I_{n}$ becomes
\begin{align}
I_{n} = \cos\left( \frac{\pi}{4n} \right) \, \Gamma\left( \frac{1}{2n} + 1 \right).
\end{align}
For the case of $n=1$ the result is
\begin{align}
\int_{0}^{\infty} \cos(t^{2}) \, dt = \frac{1}{2} \sqrt{\frac{\pi}{2}}.
\end{align}