# Solve double integral using change of variables [closed]

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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$$\int_0^3 \int_{y^2}^9 y \cos(x^2) dx dy=\int_{0}^{\sqrt{x}}\int_{0}^9y \cos(x^2)dxdy=\int_0^9\dfrac{1}{2}x\cos (x^2)dx=\dfrac{\sin 81}{4}.$$
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.