I am currently learning about Jacobians, and I need help on the following integral:
$$ \int_0^3 \int_{y^2}^9 y \cos(x^2) dx dy. $$
I am currently learning about Jacobians, and I need help on the following integral:
$$ \int_0^3 \int_{y^2}^9 y \cos(x^2) dx dy. $$
$$ \int_0^3 \int_{y^2}^9 y \cos(x^2) dx dy=\int_{0}^{\sqrt{x}}\int_{0}^9y \cos(x^2)dxdy=\int_0^9\dfrac{1}{2}x\cos (x^2)dx=\dfrac{\sin 81}{4}. $$
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.