10 questions linked to/from Some way to integrate $\sin(x^2)$?
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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 ...
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Prove: $\int_0^\infty \sin (x^2) \, dx$ converges.

$\sin x^2$ does not converge as $x \to \infty$, yet its integral from $0$ to $\infty$ does. I'm trying to understand why and would like some help in working towards a formal proof.
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In regards to the above proof, I'm a little confused as to how the last conclusion was made -- How does the fact that $$\int_{-\infty}^{\infty}e^{-x^2}dx = \sqrt{\pi}$$ to conclude that: $$\int_0^{... 4answers 278 views Find \int_{0}^{\infty} \sin x^{2}\,dx This problem is from my textbook of complex analysis. I have attempted this as: let$$u=x^{2}$$then$$dx=\frac{du}{2\sqrt{u}}$$therefore$$\frac{1}{2}\int_{0}^{\infty} \frac{\sin u}{\sqrt{u}}\,du $$... 1answer 1k views Fresnel Integrals via Differentiation under the Integral Sign I've been trying to compute \int_{-\infty}^{\infty}sin( x^2)dx via the feynman method with no luck. I was able to compute the Gaussian integral. The trick failed for fresnel integrals. Any ... 1answer 276 views How to integrate \sin (x^2) function? [closed] what are the steps to integrate the following - Not numerical method , but the integrated function is required.$$ \int \sin(x^2)\,dx.$$2answers 110 views \lim\limits_{t \to \infty} f'(t) = 0 \impliedby \lim\limits_{t \to \infty} f(t)=c  is not true! Suppose f(t) is continuously differentiable and c is a finite constant. We know that \lim\limits_{t \to \infty} f(t)=c \implies\lim\limits_{t \to \infty} f'(t) = 0 \quad \text{... 0answers 423 views Integral of \sin(x^2) From 0 to Infinity In my complex analysis final yesterday we computed$$\int_{-\infty}^\infty{\sin(x^2)dx}=\int_{-\infty}^\infty{\cos(x^2)dx}=\sqrt{\pi/2}$$using contour integration. It seems similar to the Gaussian$$\...
Do you mind expanding on the part along the diagonal in the first answer by Robjohn to this question proof? Particularly how to achieve (3). I am trying use a parameterization for $z=xe^{i \pi/4}$ ...