Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?

I have seen the Fresnel integral

$$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$

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: $$ \int_0^{\infty}\cos(\alpha t)e^{-\lambda t}\,\mathrm{d}t=\frac{\lambda}{\alpha^2+\lambda^2}\tag{1} $$ We will also use the standard arctangent integral: $$ \int_0^\infty\frac{\mathrm{d}t}{a^2+t^2}=\frac\pi{2a}\tag{2} $$ Now $$ \begin{align} &\left(\int_0^\infty\color{#C00000}{\sin}(x^2) e^{-\lambda x^2}\,\mathrm{d}x\right)^2\\ &=\int_0^\infty\int_0^\infty \color{#C00000}{\sin}(x^2)\color{#C00000}{\sin}(y^2) e^{-\lambda(x^2+y^2)}\,\mathrm{d}y\,\mathrm{d}x\tag{3.1}\\ &=\frac12\int_0^\infty\int_0^\infty \left(\cos(x^2-y^2) \color{#FF0000}{-}\cos(x^2+y^2)\right) e^{-\lambda(x^2+y^2)}\,\mathrm{d}y\,\mathrm{d}x\tag{3.2}\\ &=\frac12\int_0^{\pi/2}\int_0^\infty \left(\cos(r^2\cos(2\phi)) \color{#FF0000}{-}\cos(r^2)\right)e^{-\lambda r^2} \,r\,\mathrm{d}r\,\mathrm{d}\phi\tag{3.3}\\ &=\frac14\int_0^{\pi/2}\int_0^\infty \left(\cos(s\cos(2\phi)) \color{#FF0000}{-}\cos(s)\right) e^{-\lambda s} \,\mathrm{d}s\,\mathrm{d}\phi\tag{3.4}\\ &=\frac14\int_0^{\pi/2}\left(\frac{\lambda}{\cos^2(2\phi)+\lambda^2} \color{#FF0000}{-}\frac{\lambda}{1+\lambda^2}\right)\,\mathrm{d}\phi\tag{3.5}\\ &=\frac12\int_0^{\pi/4} \frac{\lambda}{\cos^2(2\phi)+\lambda^2}\,\mathrm{d}\phi \color{#FF0000}{-}\frac{\lambda\pi/8}{1+\lambda^2}\tag{3.6}\\ &=\frac14\int_0^{\pi/4} \frac{\lambda\,\mathrm{d}\tan(2\phi)} {1+\lambda^2+\lambda^2\tan^2(2\phi)} \color{#FF0000}{-}\frac{\lambda\pi/8}{1+\lambda^2}\tag{3.7}\\ &=\frac14\int_0^\infty\frac{\mathrm{d}t}{1+\lambda^2+t^2} \color{#FF0000}{-}\frac{\lambda\pi/8}{1+\lambda^2}\tag{3.8}\\ &=\frac{\pi/8}{\sqrt{1+\lambda^2}} \color{#FF0000}{-}\frac{\lambda\pi/8}{1+\lambda^2}\tag{3.9} \end{align} $$

$(3.1)$ 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 $$ \int_0^\infty\color{#C00000}{\sin}(x^2)\,\mathrm{d}x=\sqrt{\frac\pi8}\tag{4} $$

Addendum

I just noticed that the same proof works for $$ \int_0^\infty\cos(x^2)\,\mathrm{d}x=\sqrt{\frac\pi8}\tag{5} $$ if each red $\color{#C00000}{\sin}$ is changed to $\cos$ and each red $\color{#FF0000}{-}$ sign is changed to $+$.

Let $u=x^2$, then $$ \int_0^\infty \sin(u) \frac{\mathrm{d} u}{2 \sqrt{u}} $$ The real analysis way of evaluating this integral is to consider a parametric family: $$\begin{eqnarray} I(\epsilon) &=& \int_0^\infty \frac{\sin(u)}{2 \sqrt{u}} \mathrm{e}^{-\epsilon u} \mathrm{d} u = \frac{1}{2} \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}\int_0^\infty u^{2n+\frac{1}{2}} \mathrm{e}^{-\epsilon u} \mathrm{d} u \\ &=& \frac{1}{2} \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} \Gamma\left(2n+\frac{3}{2}\right) \epsilon^{-\frac{3}{2}-2n} \\ &=& \frac{1}{2 \epsilon^{3/2}} \sum_{n=0}^\infty \left(-\frac{1}{\epsilon^2}\right)^n\frac{\Gamma\left(2n+\frac{3}{2}\right)}{\Gamma\left(2n+2\right)} \\ &\stackrel{\Gamma-\text{duplication}}{=}&\frac{1}{2 \epsilon^{3/2}} \sum_{n=0}^\infty \left(-\frac{1}{\epsilon^2}\right)^n\frac{\Gamma\left(n+\frac{3}{4}\right)\Gamma\left(n+\frac{5}{4}\right)}{\sqrt{2} n! \Gamma\left(n+\frac{3}{2}\right)} \\ &=& \frac{1}{(2 \epsilon)^{3/2}} \frac{\Gamma\left(\frac{3}{4}\right)\Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{3}{2}\right)} {}_2F_1\left(\frac{3}{4}, \frac{5}{4}; \frac{3}{2}; -\frac{1}{\epsilon^2}\right) \\ &\stackrel{\text{Euler integral}}{=}& \frac{1}{(2 \epsilon)^{3/2}} \frac{\Gamma\left(\frac{3}{4}\right)\Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{3}{2}\right)} \frac{1}{\operatorname{B}\left(\frac{5}{4}, \frac{3}{2}-\frac{5}{4}\right)} \int_0^1 x^{\frac{5}{4}-1} (1-x)^{\frac{3}{2}-\frac{5}{4} -1} \left(1+\frac{x}{\epsilon^2}\right)^{-3/4} \mathrm{d} x \\ &=& \frac{1}{2^{3/2}} \frac{\Gamma\left(\frac{3}{4}\right)\Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{3}{2}\right)} \frac{\Gamma\left(\frac{3}{2}\right)}{\Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{1}{4}\right)} \int_0^1 x^{\frac{5}{4}-1} (1-x)^{\frac{1}{4} -1} \left(\epsilon^2+x\right)^{-3/4} \mathrm{d} x \end{eqnarray} $$ Now we are ready to compute $\lim_{\epsilon \to 0} I(\epsilon)$: $$\begin{eqnarray} \lim_{\epsilon \to 0} I(\epsilon) &=& \frac{1}{2^{3/2}} \frac{\Gamma\left(\frac{3}{4}\right)}{\Gamma\left(\frac{1}{4}\right)} \int_0^1 x^{\frac{1}{2}-1} \left(1-x\right)^{\frac{1}{4}-1} \mathrm{d} x = \frac{1}{2^{3/2}} \frac{\Gamma\left(\frac{3}{4}\right)}{\Gamma\left(\frac{1}{4}\right)} \frac{\Gamma\left(\frac{1}{2}\right) \Gamma\left(\frac{1}{4}\right)}{\Gamma\left(\frac{3}{4}\right)} \\ &=& \frac{1}{2^{3/2}} \Gamma\left(\frac{1}{2}\right) = \frac{1}{2} \sqrt{\frac{\pi}{2}} \end{eqnarray} $$

First, use the substitution $t = x^2$ to obtain

$$ I := \int_{0}^{\infty}\sin\left(x^2\right)\;dx=\int_{0}^{\infty}\frac{\sin t}{2\sqrt{t}}\;dt. $$

This integral converges conditionally, thus we make an integration by parts to obtain an absolutely convergent integral as follows:

$$I = \left[\frac{1-\cos t}{2\sqrt{t}}\right]_{0}^{\infty}-\int_{0}^{\infty}\left(\frac{d}{dt}\frac{1}{2\sqrt{t}}\right)(1 - \cos t)\;dt =\int_{0}^{\infty}\frac{1 - \cos t}{4t^{3/2}}\;dt.$$

Now, from gamma integral,

$$ \begin{align*} I &=\frac{1}{4\Gamma(3/2)}\int_{0}^{\infty}\left(\frac{\Gamma(3/2)}{t^{3/2}}\right)(1 - \cos t)\;dt \\ &=\frac{1}{4\Gamma(3/2)}\int_{0}^{\infty}\left(\int_{0}^{\infty}u^{1/2}e^{-tu}\;du\right)(1 - \cos t)\;dt\\ &=\frac{1}{4\Gamma(3/2)}\int_{0}^{\infty}\int_{0}^{\infty}u^{1/2}e^{-tu}(1 - \cos t)\;dudt \\ &=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\int_{0}^{\infty}u^{1/2}e^{-tu}(1 - \cos t)\;dudt, \end{align*} $$

where in the last line we have used the fact that $\Gamma(3/2) = \frac{1}{2}\sqrt{\pi}$, which is a direct consequence of the Gaussian integral. (Of course, this integral can be evaluated by a famous real analysis technique.) By Tonelli's theorem we can change the order of integration, and with the substitution $u = v^2$ we obtain

$$\begin{align*}I &=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\int_{0}^{\infty}u^{1/2}e^{-tu}(1 - \cos t)\;dtdu \\ &= \frac{1}{2\sqrt{\pi}} \int_{0}^{\infty}\frac{u^{1/2}}{u(1+u^2)}\;du = \frac{1}{2\sqrt{\pi}} \int_{0}^{\infty} \frac{2}{1+v^4} \; dv. \end{align*}$$

To evaluate the last integral, we use the following decomposition

$$ \frac{2}{1+v^4} = \frac{1+v^{-2}}{(v-v^{-1})^2-2} - \frac{1-v^{-2}}{(v+v^{-1})^2-2}. $$

Thus with the substitution $z = v - v^{-1}$ and $w = v + v^{-1}$, the integral becmes

$$\begin{align*}I &= \frac{1}{2\sqrt{\pi}} \left( \int_{0}^{\infty} \frac{d(v-v^{-1})}{(v-v^{-1})^2+2} - \int_{0}^{\infty} \frac{d(v+v^{-1})}{(v+v^{-1})^2-2} \right) \\ &= \frac{1}{2\sqrt{\pi}} \left( \int_{-\infty}^{\infty} \frac{dz}{z^2+2} - \int_{\infty}^{\infty} \frac{dw}{w^2-2} \right) = \frac{1}{2\sqrt{\pi}} \left( \frac{\pi}{\sqrt{2}} - 0 \right) = \sqrt{\frac{\pi}{8}}. \end{align*}$$

Only a slight modification of this argument immediately yields

$$ \int_{0}^{\infty} \cos\left(x^2\right) \; dx = \sqrt{\frac{\pi}{8}}. $$

Everybody who has not at least quick-browse throught some pages of Integral Kokeboken brochure (although in norwegian), I definitelly recommend to do so. In this gem one can find : (for improving redability I will adopt the methodology used by @robjohn, I will even use the same apriori identities)

$$ \begin{align} &\int_0^\infty\sin{(x^2)}\;\mathrm{d}x=\\ &=\frac{1}{2}\int_0^\infty\sin{u}\;\frac{\mathrm{d}u}{\sqrt{u}}\tag{1.1}\\ &=\frac{1}{2}\int_0^\infty\sin{u}\;\frac{2}{\sqrt{\pi}}\int_{0}^\infty e^{-uv^2}\;\mathrm{d}v\,\mathrm{d}u\tag{1.2} \\ &=\frac{1}{\sqrt{\pi}}\int_0^\infty\int_{0}^\infty e^{-uv^2}\sin{u}\;\mathrm{d}u \,\mathrm{d}v\tag{1.3}\\ &=\frac{1}{\sqrt{\pi}}\int_0^\infty\frac{\mathrm{d}v}{1+v^4}\tag{1.4}\\ &=\frac{1}{2\sqrt{\pi}}\int_0^\infty\frac{1+v^2}{1+v^4}\;\mathrm{d}v\tag{1.5}\\ &=\frac{1}{2\sqrt{\pi}}\int_0^\infty\frac{d(v-v^{-1})}{(v-v^{-1})^2+2}\tag{1.6}\\ &=\frac{1}{2\sqrt{\pi}}\int_{-\infty}^\infty\frac{dw}{w^2+2}=\frac{1}{2\sqrt{2\pi}}\arctan\left(\frac{w}{\sqrt{2}}\right)\bigg{|}_{-\infty}^{\infty}=\sqrt{\frac{\pi}{8}}\tag{1.7}\\ \end{align} $$

Notes :

$(1.1)$ substitution $u=x^2$

$(1.2)$ Gaussian integral

$(1.3)$ changing the order of integration

$(1.4)$ the second apriori formula, we will get the same integral as @Sangchul Lee got.

$(1.5)$ same method, but slightly modified, wrtite the integral as $I=(I+I)/2$ then in the second make substitution $v\rightarrow 1/v$ to get this form

$(1.6)$ dividing both numeratpr and denominator by $v^2$, completing the differential

$(1.7)$ substitution $w=v-v^{-1}$, doesn't need any commentary :)

Similarly for $\cos(x^2)$.

Here is paper that addresses this exact question by H. Flanders, "On Fresnel integrals", American Mathematical Monthly, vol. 89, no. 4, 1982, pp. 264-266.

Here is another paper on the topic that does not require a subscription.