Generalized Fresnel integral $\int_0^\infty \sin x^p \, {\rm d}x$ I am stuck at this question. Find a closed form (that may actually contain the Gamma function) of the integral
$$\int_0^\infty \sin (x^p)\, {\rm d}x$$
I am interested in a Laplace approach, double integral etc. For some weird reason I cannot get it to work. 
I am confident that a closed form may actually exist since for the integral:
$$\int_0^\infty \cos x^a \, {\rm d}x = \frac{\pi \csc \frac{\pi}{2a}}{2a \Gamma(1-a)}$$
there exists a closed form with $\Gamma$ and can be actually be reduced further down till it no contains no $\Gamma$. But trying to apply the method of Laplace transform that I have seen for this one , I cannot get it to work for the above integral that I am interested in. 
May I have a help?

Hint: Use $~\displaystyle\int_0^\infty\exp\Big(-\sqrt[n]x\Big)~dx~=~n!~\iff~\int_0^\infty e^{-x^n}~dx~=~\Gamma\bigg(1+\frac1n\bigg)~$ in conjunction with Euler's formula.
