Why is it impossible to evaluate $ \int_0^1\int_x^1 \sin ( y ^ 2 ) \, dy \, dx $ in this order?

Question

Why is it impossible to evaluate the iterated integral $$ \int _ 0 ^ 1 \int _ x ^ 1 \sin ( y ^ 2 ) \, d y \, d x $$ in this order? I only was able to solve it after changing the order of integration.

Answer 1

Suppose that we do not change the order of integration. $$\int\sin ( y ^ 2 ) \, d y=\sqrt{\frac{\pi }{2}} S\left(\sqrt{\frac{2}{\pi }} y\right)$$ $$\int _ x ^ 1 \sin ( y ^ 2 ) \, d y=\sqrt{\frac{\pi }{2}} S\left(\sqrt{\frac{2}{\pi }}\right)-\sqrt{\frac{\pi }{2}} S\left(\sqrt{\frac{2}{\pi }} x\right) $$ Now, using one integration by parts, $$\int S\left(\sqrt{\frac{2}{\pi }} x\right)\,dx=x S\left(\sqrt{\frac{2}{\pi }} x\right)+\frac{\cos \left(x^2\right)}{\sqrt{2 \pi }}$$ $$\int_0^1 S\left(\sqrt{\frac{2}{\pi }} x\right)\,dx=S\left(\sqrt{\frac{2}{\pi }}\right)-\frac{1}{\sqrt{2 \pi }}+\frac{\cos (1)}{\sqrt{2 \pi }}$$ and all of that to arrive at $$\int _ 0 ^ 1 \int _ x ^ 1 \sin ( y ^ 2 ) \, d y \, d x=\frac{1-\cos (1)}{2}$$

What do you prefer (even knowing everything about Fresnel integrals) ?