I was working on a Calculus Challenge worksheet and came across this question: Determine whether the improper integral $\int \limits_{0}^{\infty}\cos (x^2) dx$ converges or diverges, and if it converges, find what value it converges to. I am completely stuck, I tried some algebraic manipulations of $x^2$ and $\cos(x)$, but nothing worked out. Please help. Also, if my formatting sucks, feel free to comment and edit the post.
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This is called the Fresnel integral, and there are lots of sources online solving this integral. See, for example, the Wikipedia page. – ljfirth Jun 18 '24 at 17:16
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https://math.stackexchange.com/q/3273031/42969, https://math.stackexchange.com/q/4003529/42969, https://math.stackexchange.com/q/187729/42969 – Martin R Jun 18 '24 at 17:18
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The Fresnel integral can be done by contour integral, but it is is bit tricky. – GEdgar Jun 18 '24 at 17:24
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@GEdgar is likely advocating this choice of contour. – J.G. Jun 18 '24 at 17:31