Editing Fubini's theorem (section)

=== Arcsine integral ===

The arcsine integral, also called the inverse sine integral, is a function that cannot be represented by [[elementary function]]s. However, some of the values of the arcsine integral can be expressed with elementary functions. These values can be determined by integrating the derivative of the arcsine integral, which is the quotient of the arcsine divided by the [[identity function]] - the cardinalized arcsine. {{Clarify|date=February 2025|reason= Isn't this just the definition of the "arcsine integral" (by analogy with the sine integral)?|text=The arcsine integral is exactly the original antiderivative of the cardinalized arcsine.}} To integrate this function, Fubini's theorem serves as a key, which unlocks the integral by exchanging the order of the integration parameters. When applied correctly, Fubini's theorem leads directly to an antiderivative function that can be integrated in an elementary way, which is shown in cyan in the following equation chain:

<math display="block">\operatorname{Si}_{2}(1) = \int_{0}^{1} \frac{1}{x}\arcsin(x) \,\mathrm{d}x = {\color{blue}\int_{0}^{1}} {\color{green}\int_{0}^{1}} \frac{\sqrt{1-x^2}\,y}{(1- x^2 y^2)\sqrt{1-y^2}} \,{\color{green}\mathrm{d}y} \,{\color{blue}\mathrm{d}x} =</math>
<math display="block">= {\color{green}\int_{0}^{1}} {\color{blue}\int_{0}^{1}} \frac{\sqrt{1-x^2}\,y}{(1-x^2 y^2)\sqrt{1-y^2}} {\color{blue}\,\mathrm{d}x} {\color{green}\,\mathrm{ d}y} =\int_{0}^{1} \frac{\pi\,y}{2\sqrt{1-y^2}(1+\sqrt{1-y^2}\,)} \,\mathrm{d}y =</math>
<math display="block">= {\color{RoyalBlue}\biggl\{ \frac{\pi}{2} \ln\bigl[2 \bigl(1 + \sqrt{1 - y^2}\,\bigr)^ {-1}\bigr] \biggr\}_{y = 0}^{y = 1}} = \frac{\pi}{2}\ln(2)</math>