The following integral has been proposed:
$$ \int_{0}^{\frac{\pi}{2}} \sqrt[8]{x^{2}+\ln ^{2} \cos x} \sqrt{\frac{1}{2}+\frac{1}{2} \sqrt{\frac{1}{2}+\frac{1}{2} \sqrt{\frac{\ln ^{2} \cos x}{x^{2}+\ln ^{2} \cos x}}}} d x=\frac{\pi}{2} \sqrt[4]{\ln 2} $$
I approximated the behavior of the LHS by summing consecutive terms (everything after the integral sign) by increasing x. This was done in Javascript.
Running x to large values shows the LHS as divergent:
...while the RHS is equal to exactly 1.307.
Presumably, at some early point, the LHS should equal the RHS. Is it feasible to discover where this is using my approach? Summation is increasing x by units of 1, which in a sense is course-graining what would otherwise be done using proper integration. But should I expect to see the “convergence” (overlap?) somewhere early in my summation?
Obviously I would need to inspect the “series” (calling it series since this is how I’m treating it) between the values of 0 and π/2. But Since I am increasing x in the regular number line I need to convert the polar coordinates to reals (I’m assuming).
Looking at the behavior of LHS in my graph we see the value 1.136 at 2 (which is the real value of π/2 ?
Is my approach correct? Is this how I can confirm the expression computationally?