Solve $\int_{0}^{\infty} a^{ix^2}$ dx with contour integration?

Question

Evaluate $$\int_{0}^{\infty} a^{ix^2} dx$$ for $0<a<1$ using contour integration methods.

Answer 1

$$a^{ix^2}=\left(e^{\ln a}\right)^{ix^2}$$ so you have: $$I=\int_0^\infty e^{i\ln(a)x^2}dx$$ then I'd suggest using the substitution $u=\sqrt{\ln(a)}x$