$\int_0^{\infty} \sin (a^2 x^2)dx$ picking the contour of integration

Question

I just have a question about what contour I should pick for this problem $$\int_0^{\infty} \sin (a^2 x^2)dx$$ for positive a.

I used the freshnel integral for $\int_0^{\infty} \sin (x^2)dx$ with an angle of $\pi/4$. (See below)

My question is, would the angle that I would use change because of the $a^2$ addition to the problem or would I want to use the same contour and just carry the $a^2$ through the problem?

Or perhaps there is a way to take care of it right off the bat? For example of what I mean, see this post.

Thanks for your time.

Answer 1

Substitute $u=ax$:

$$ \int_0^\infty\sin\left(a^2x^2\right)\mathrm dx=\frac1a\int_0^\infty\sin\left(a^2x^2\right)\mathrm d(ax)=\frac1a\int_0^\infty\sin u^2\mathrm du\;. $$