Integration: TI-86 & Casio Algebra 2.0 Plus

[carey]

Valentin’s excellent set of 2024 PI day examples for vintage HP calculators includes in the 1st example the integral \( \int_{0}^{1} \frac{tan^{-1} tanh^{-1} x}{x} dx \). When attempting this integral on two of my favorite non-hp calculators, the TI-86 and the Casio Algebra 2.0 Plus, I found it interesting that the TI-86 proceeded (albeit slowly) to an answer (screenshot 1) but the Casio Algebra 2.0 Plus gave an “Ma Error” (math error) if either the lower limit was 0 or, independently, if the upper limit was 1 (only one screenshot shown) and it was necessary to back away slightly from both limits of integration (0 and 1, substituting, for example, 0.001 and 0.999) to obtain a result.

My guess is that the Casio “freaked out” (to use technical language :) at the lower integration limit of 0 because of apparent division by 0 and ignored that the integrand approaches a limit of 1 at x=0. My guess for the Casio balking at the upper integration limit of 1 is that it’s a cusp, and ignored that the integrand approaches a limit of \( \frac{\pi}{2} \) at x=1.

I’d be interested in hearing your thoughts about this discrepancy between the numerical integration methods and results for these two calculators. Thanks!