I have to prove that for $f(x,y)=e^{-xy^2}\sin(x)$ and $\forall b>0$ we have
$$\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$$
I've tried to solve the first integral but I'm only arriving to indeterminations...
Then, I have to prove that
$$\lim_{b\to \infty} \int_{0}^b \dfrac{\sin x}{\sqrt{x}} = \lim_{b\to \infty} \int_{0}^b \dfrac{\cos x}{\sqrt{x}}=\sqrt{\dfrac{\pi}{2}} $$.
I suppose that's realated with the results of the first integrals and I will be able to solve it when I finally prove the first equation.
Any tips please?
First we note I, II and III.
I. For positive $x$, $$\int_{0}^\infty e^{-xy^2}\sin x \,dy=\frac{\sqrt {\pi}}{2}\cdot\frac{\sin x}{\sqrt{x}}.$$ Use $\int_0^{\infty} e^{-y^2} dy$=$\frac{\sqrt\pi}{2}$.
II. Integration by parts leads to$$\int e^{-xy^2}\sin x dx=-\frac{e^{-xy^2}}{1+y^4} (\cos x+y^2\sin x) .$$
III. For all $a, b$ with $0<a<b<\infty$,$$\int_a^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{a}^b f \,dx\right) dy .\tag{1}$$ Since $|e^{-xy^2}\sin x|\le e^{-ay^2}$ for all $ x (a\le x<\infty)$ and $\int_0^{\infty} e^{-ay^2}\,dy<\infty $ ,we can change the order of integration.
Now we prove that for any $b>0$,$$\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy .\tag{2}$$ Proof. We have $$\int_0^b \left(\int_{0}^\infty f \,dy\right) dx=\int_0^a \left(\int_{0}^\infty f \,dy\right) dx+\int_a^b \left(\int_{0}^\infty f \,dy\right) dx .\tag{3}$$The first term of the right hand side$$\int_0^a \left(\int_{0}^\infty f \,dy\right) dx=\int_0^a\frac{\sqrt{\pi}}{2}\cdot\frac{\sin x}{\sqrt{x}} dx\to 0 $$as $a\to 0$. Simirarly we have $$ \int_0^\infty \left(\int_{0}^b f \,dx\right) dy= \int_0^\infty \left(\int_{0}^a f \,dx\right) dy+ \int_0^\infty \left(\int_a^b f \,dx\right) dy\tag{4}$$ and the first term of the right hand side $$\int_0^\infty \left(\int_{0}^a f \,dx\right) dy =\int_0^\infty \left( \frac{1-e^{-ay^2}\cos a}{1+y^4}- \frac{y^2e^{-ay^2}\sin a}{1+y^4}\right) dy $$tends to $0$ as $a\to 0$. Therefore, tending $a$ to $0$ in (3) and (4), we have (2) due to (1).
Second question:
From (2),we have $$\frac{\sqrt{\pi}}{2}\int_0^b \frac{\sin x}{\sqrt{x}} dx =\int_0^{\infty} \left(\frac{1}{1+y^4}-\frac{e^{-by^2}}{1+y^4}(\cos b+y^2\sin b)\right) dy .$$ Thus we have $$\frac{\sqrt{\pi}}{2}\int_0^\infty \frac{\sin x}{\sqrt{x}} dx =\int_0^{\infty} \frac{1}{1+y^4} dy $$ and $$\int_0^\infty \frac{\sin x}{\sqrt{x}} dx =\sqrt{\dfrac{\pi}{2}}.$$
The same argument for $f(x,y)=e^{-xy^2}\cos x$ leads to $$\int_{0}^\infty \dfrac{\cos x}{\sqrt{x}}=\sqrt{\dfrac{\pi}{2}}.$$