$\int_{0}^{\infty}{\cos({x^2})\,dx}$
- Grigorios Kostakos
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$\int_{0}^{\infty}{\cos({x^2})\,dx}$
Calculate the integral:
$$\int_{0}^{\infty}{\cos({x^2})\,dx}\,.$$
$$\int_{0}^{\infty}{\cos({x^2})\,dx}\,.$$
Grigorios Kostakos
- Tolaso J Kos
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Re: $\int_{0}^{\infty}{\cos({x^2})\,dx}$
We have that:
$$\int_{0}^{\infty}\cos x^2 \, {\rm d}x = \mathfrak{Re}\left ( \int_{0}^{\infty}e^{-ix^2}\, {\rm d}x \right )= \frac{1}{2}\sqrt{\frac{\pi}{2}}$$
The latter integral has been proved here.
Note, also, that:
$$\int_{0}^{\infty}\sin x^2 \, {\rm d}x = \mathfrak{Im}\left ( \int_{0}^{\infty}e^{-ix^2}\, {\rm d}x \right )= \frac{1}{2}\sqrt{\frac{\pi}{2}}$$
$$\int_{0}^{\infty}\cos x^2 \, {\rm d}x = \mathfrak{Re}\left ( \int_{0}^{\infty}e^{-ix^2}\, {\rm d}x \right )= \frac{1}{2}\sqrt{\frac{\pi}{2}}$$
The latter integral has been proved here.
Note, also, that:
$$\int_{0}^{\infty}\sin x^2 \, {\rm d}x = \mathfrak{Im}\left ( \int_{0}^{\infty}e^{-ix^2}\, {\rm d}x \right )= \frac{1}{2}\sqrt{\frac{\pi}{2}}$$
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