Some way to integrate $\sin(x^2)$? [duplicate]

Because the straight forward approach involves Fresnel integrals I thought about a different approach of taking the imaginary part of $\int_{-\infty}^{\infty}\exp{(ix^2)} \, dx $ but have no idea how to continue.

Knowing that $\int_{-\infty}^{\infty}\exp{(-x^2)} \, dx = \sqrt \pi$ one can feel that this value has to be somehow "divided" over the entire complex plane, so that every quadrant gets some part of $\sqrt \pi$. This would give the desired result of $\sqrt \frac{\pi}{2} - i \sqrt \frac{\pi}{2} $

Is there a way this could be proved or justified?

I think robjohn's answer is excellent, but I also think it would be helpful to provide the details explaining why the integral along the arc vanishes as $R \to \infty$.

For the path along the arc we may choose $\gamma(t) = R e^{i\frac{\pi}{4}t}$ for $0 \leq t \leq 1$. We will show $$\lim_{R \to \infty} \bigg| \int_\gamma e^{-z^2} dz \bigg| = 0.$$ For fixed $R$, we have $$\bigg| \int_\gamma e^{-z^2} dz \bigg| \leq \int_\gamma \bigg| e^{-z^2} \bigg| d|z| = \int_0^1 \bigg| e^{-[\gamma(t)]^2} \bigg| \big| \gamma^\prime(t) \big| dt = R \frac{\pi}{4} \int_0^1 \bigg| e^{-R^2 e^{i \frac{\pi}{2} t}} \bigg| dt. $$ Decomposing the exponential in the numerator, this becomes $$R \frac{\pi}{4} \int_0^1 e^{-R^2 \cos(\frac{\pi}{2}t)} dt \leq R \frac{\pi}{4} \int_0^1 e^{-R^2(1-t)} dt = R \frac{\pi}{4} \int_0^1 e^{-R^2 u} du = \frac{\pi}{4} \frac{1}{R} (1 - e^{-R^2}). $$ Hence, $$\lim_{R \to \infty} \bigg| \int_\gamma e^{-z^2} dz \bigg| \leq 0, $$ as desired.

Check out this answer for a real method to evaluate this integral.

First note that $$ \int_0^\infty e^{-x^2}\,\mathrm{d}x=\frac{\sqrt\pi}2\tag{1} $$

The integral $$ \int_\gamma e^{-z^2}\,\mathrm{d}z=0\tag{2} $$ over the curve $\gamma$ consisting of the line from $0$ to $R$ then counterclockwise along the circular arc from $R$ to $Re^{i\pi/4}$ then back along the line from $Re^{i\pi/4}$ to $0$ must be $0$ since $e^{-z^2}$ has no singularities inside $\gamma$.

Next note that $\int_\gamma e^{-z^2}\,\mathrm{d}z$ along the line from $0$ to $R$ as $R\to\infty$ tends to $(1)$.

Next note that $\int_\gamma e^{-z^2}\,\mathrm{d}z$ along the arc of the circle of radius $R$ from $R$ to $Re^{i\pi/4}$ as $R\to\infty$ goes to zero.

Finally, note that $\int_\gamma e^{-z^2}\,\mathrm{d}z$ along the line from $Re^{i\pi/4}$ to $0$ is $$ -e^{i\pi/4}\int_0^\infty e^{-ix^2}\,\mathrm{d}x\tag{3} $$ Therefore, $(1)$ plus $(3)$ is $0$, so we get $$ \int_0^\infty e^{-ix^2}\,\mathrm{d}x=\frac{1-i}{\sqrt2}\frac{\sqrt\pi}2\tag{4} $$ Taking the imaginary part of $(4)$ yields that $$ \int_0^\infty\sin(x^2)\,\mathrm{d}x=\sqrt{\frac\pi8}\tag{5} $$ or $$ \int_{-\infty}^\infty\sin(x^2)\,\mathrm{d}x=\sqrt{\frac\pi2}\tag{6} $$

Starting from $\int_{-\infty}^\infty \exp(-x^2)\ dx = \sqrt{\pi}$, use a change of variables to get $\int_{-\infty}^\infty \exp(-ax^2)\ dx = \sqrt{\pi/a}$ for $a > 0$. Now both sides are analytic in $a$ for $\text{Re}(a) > 0$ (using a branch of the square root that is analytic in the right half plane), so the equation should still be true for $\text{Re}(a) > 0$. The next part is tricky to justify: you want to take the limit as $a \to -i$. Assuming this is valid, you have

$$ \int_{-\infty}^\infty \exp(ix^2)\ dx = \sqrt{\frac{\pi}{-i}}= (1+i) \sqrt{\frac{\pi}{2}}$$

so that $\int_{-\infty}^\infty \cos(x^2)\ dx = \int_{-\infty}^\infty \sin(x^2)\ dx = \sqrt{\pi/2}$.