# 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:

Now

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

$(3.2)$ use $2\color{#C00000}{\sin}(A)\color{#C00000}{\sin}(B)=\cos(A-B)\color{#FF0000}{-}\cos(A+B)$

$(3.3)$ convert to polar coordinates

$(3.4)$ substitute $s=r^2$

$(3.5)$ apply $(1)$

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

$(3.7)$ multiply numerator and denominator by $\sec^2(2\phi)$

$(3.8)$ substitute $t=\lambda\tan(2\phi)$

$(3.9)$ apply $(2)$

Finally, take the square root of both sides of $(3)$ and let $\lambda\to0^+$ to get

if each red $\color{#C00000}{\sin}$ is changed to $\cos$ and each red $\color{#FF0000}{-}$ sign is changed to $+$.