Evaluating ∫∞0sinx2dx\int_0^\infty \sin x^2\, dx with real methods?

I have seen the Fresnel integral


evaluated by contour integration and other complex analysis methods, and I have found these methods to be the standard way to evaluate this integral. I was wondering, however, does anyone know a real analysis method to evaluate this integral?


Aside from some trigonometric substitutions and identities, we will need the following identity, which can be shown using integration by parts twice:
We will also use the standard arctangent integral:

(3.1) change the square of the integral into a double integral

(3.2) use 2sin(A)sin(B)=cos(AB)cos(A+B)

(3.3) convert to polar coordinates

(3.4) substitute s=r2

(3.5) apply (1)

(3.6) pull out the constant and apply symmetry to reduce the domain of integration

(3.7) multiply numerator and denominator by sec2(2ϕ)

(3.8) substitute t=λtan(2ϕ)

(3.9) apply (2)

Finally, take the square root of both sides of (3) and let λ0+ to get


I just noticed that the same proof works for
if each red sin is changed to cos and each red sign is changed to +.

Source : Link , Question Author : Argon , Answer Author : robjohn

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