$\int_{0}^{\infty}{\cos({x^2})\,dx}$

Calculus (Integrals, Series)
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Grigorios Kostakos
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$\int_{0}^{\infty}{\cos({x^2})\,dx}$

#1

Post by Grigorios Kostakos »

Calculate the integral:

$$\int_{0}^{\infty}{\cos({x^2})\,dx}\,.$$
Grigorios Kostakos
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Tolaso J Kos
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Re: $\int_{0}^{\infty}{\cos({x^2})\,dx}$

#2

Post by Tolaso J Kos »

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:

$$\int_{0}^{\infty}\sin x^2 \, {\rm d}x = \mathfrak{Im}\left ( \int_{0}^{\infty}e^{-ix^2}\, {\rm d}x \right )= \frac{1}{2}\sqrt{\frac{\pi}{2}}$$
Imagination is much more important than knowledge.
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