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### 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 ...
247 views

### How to calculate $\int_{0}^{+\infty} {\sin {x^2}\mathrm{d}x}$? [duplicate]

Possible Duplicate: Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods? How to calculate $\displaystyle\int_{0}^{+\infty} {\sin {x^2}\mathrm{d}x}$ ?
164 views

### Evaluate $\int_{0}^{\infty} \cos(x^2)dx$ [duplicate]

Prove that the above integral is equal to $\frac{\sqrt{2\pi}}{2}$ I have already tried expanding using $\cos$ identity and also taking Laplace for it. I am getting nowhere with this.
45k views

### Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$

How to prove $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
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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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### Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$

Numerically it seems to be true that $$\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}.$$ Any ideas how to prove this?
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### Generalizing the trick for integrating $\int_{-\infty}^\infty e^{-x^2}\mathrm dx$?

There is a well-known trick for integrating $\int_{-\infty}^\infty e^{-x^2}\mathrm dx$, which is to write it as $\sqrt{\int_{-\infty}^\infty e^{-x^2}\mathrm dx\int_{-\infty}^\infty e^{-y^2}\mathrm dy}$...
### tough integral involving $\sin(x^2)$ and $\sinh^2 (x)$
I ran across this integral I get no where with. Can someone suggest a method of attack?. $$\int_0^{\infty}\frac{\sin(\pi x^2)}{\sinh^2 (\pi x)}\mathrm dx=\frac{2-\sqrt{2}}{4}$$ I tried series, ...