33k views

### 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 ...
528 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.
346 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 $\displaystyle\int_{0}^{+\infty} {\sin {x^2}\mathrm{d}x}$ ?
152k views

### Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$

How to prove $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
33k views

### 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.
8k views

### 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?