I'd like to calculate $\int_{-\infty}^{\infty}\sin(x^2)dx$ and $\int_{-\infty}^{\infty}\cos(x^2)dx$ I think it may be possible to do it by using the fact that: $$\int_{\gamma}^{} e^{-z^2}dz =0$$ For any closed curve $\gamma$ and that $$\int_{-\infty}^{+\infty}e^{-x^2}dx=\pi^{\frac{1}{2}}$$
but I can't finish it , because I am not able to find a proper curve.How can I finish it , or is there another way to do it? Will be thankful for any clues.