I'm trying to evaluate the following integral: $$\int_{0}^{1} \int_{0}^{\pi/2} \frac{\cos{y} \ln{xy}}{\sqrt{x}} dy dx$$
Now this is an improper double integral, and I am unsure as to how to use limits like we do when dealing with improper integrals (e.g. replace the part of the interval causing the issue with a dummy variable, and taking the limit as the dummy variable approaches that value).
On the line $x=0$, the function blows up to infinity. Similarly, if $xy$ is less than 1, the function blows up to infinity. There are lots of discontinuities sort of tangled together, and I don't know how to deal with this rigorously using limits.
Thanks
Hint. Perform the transformation $xy=u,$ so that the integral becomes $$\int_0^{π/2}\frac{\cos y}{\sqrt y}\mathrm dy\int_0^y \frac{\log u}{\sqrt u}\mathrm du.$$ Then integrate the first integral by parts, to obtain the integral $$2\int_0^{π/2}\cos y(\log y-2)\mathrm dy.$$ Everything here is straightforward, except for the first part, which may be done by parts to obtain $\operatorname{Si}(π/2).$