I am currently learning about Jacobians, and I need help on the following integral:
$$ \int_0^3 \int_{y^2}^9 y \cos(x^2) dx dy. $$
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2$\begingroup$ hello, welcome to MSE. Please consider using mathjax: math.meta.stackexchange.com/questions/5020/… to typeset your question. It would also be helpful to know what you tried and where you're stuck. $\endgroup$– Rahul MadhavanApr 9, 2021 at 12:12
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$\begingroup$ Please do not write "please help me I need this quickly". If you really want your question to be answered more quickly, see how to ask a good question so that your question is more presentable. $\endgroup$– Toby MakApr 9, 2021 at 13:21
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2 Answers
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$$ \int_0^3 \int_{y^2}^9 y \cos(x^2) dx dy=\int_{0}^{\sqrt{x}}\int_{0}^9y \cos(x^2)dxdy=\int_0^9\dfrac{1}{2}x\cos (x^2)dx=\dfrac{\sin 81}{4}. $$
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There is no way to represent $\int\cos(x^2)dx$ in terms of elementary functions! This is called the Fresnel integral, see Fresnel Integral. The best you can do is use series expansions.
