Generalized Fresnel Integration: $\int_{0}^ {\infty } \sin(x^n) dx $ and $\int_{0}^ {\infty } \cos(x^n) dx $

Question

I calculated generalized (w.r.t. the degree) Fresnel integrals $\int_{0}^ {\infty } \sin(x^n) dx $ and $\int_{0}^ {\infty } \cos(x^n) dx $ by contour integration. And then I got $$\int_{0}^ {\infty } \sin(x^n) dx = \frac{1}{2i}(i^{1/n} - i^{-1/n}) \frac{1}{n}\Gamma\!\left(\frac{1}{n}\right), \\ \int_{0}^ {\infty } \cos(x^n) dx = \frac{1}{2}(i^{1/n} + i^{-1/n}) \frac{1}{n}\Gamma\!\left(\frac{1}{n}\right).$$ Are these formulas correct? Can anyone please check their validity?