What is the integral from 0 to infinity of ( cos x / sqrt{x} )?

Question

Give the definite integral from 0 to infinity of cos x / sqrt{x} dx. (ie the function is x^{-1/2} cos x..

Answer 1

Make the substitution x=u^2 to get 2 times the integral of cos(u^2)du from 0 to infinity. This is called a Fresnel integral and has to be evaluated via a complex integration. The final value is
2sqrt(pi/8)=sqrt(pi/2).