Contour integration: $\int_0^\infty \frac{\cos x}{\sqrt{x}}dx$

Question

A problem from a complex analysis qualifier:

Find $$\int_0^\infty \frac{\cos x}{\sqrt{x}}dx$$

My answer so far: We want to integrate the function $$f(z) = \frac{\cos z}{\sqrt{z}}dx = e^{iz-\frac12 \log z} + e^{-iz-\frac12 \log z}$$ for a branch of $\log$.

Answer 1

Hint: $\displaystyle\int_0^\infty\frac{\cos x}{\sqrt x}dx=2\int_0^\infty\frac{\cos t^2}tt\cdot dt=2\,\Re\int_0^\infty e^{it^2}dt\ -$ Can you take it from here ? :-)

Answer 2

Hint: this $$\int_{0}^{\infty}\dfrac{\cos{x}}{\sqrt{x}}dx=2\int_{0}^{\infty}\cos{(x^2)}dx=2\cdot\dfrac{\sqrt{\pi}}{2}$$