Why does the definite integral of $\frac{\cos(x)}{\sqrt{x}}$ converge

Question

I was taking the integral, $\int_0^\infty \frac{\sin(x)}{\sqrt{x}}$ and I found out in this process that the answer is equivalent to $\int_0^\infty \frac{\cos(x)}{\sqrt{x}}$ where they both equal $\sqrt{\pi/2}$. However I am very confused as to how $\int_0^\infty \frac{\cos(x)}{\sqrt{x}}$ may converge to an answer when it diverges at the boundary of x = 0.