Integration: TI-86 & Casio Algebra 2.0 Plus

[Steve Simpkin]

(03-23-2024 05:09 PM)carey Wrote:  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!

For reference, here are a few timings and results for integrating the above problem using some of the calculators I had handy. The Casio fx-991EX is the only one that gave an error if I used the limits of 0 to 1.

Model, lower limit, upper limit, timing = result
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HP Prime (simulation), 0, 1, instant = 1.02576051093
Casio fx-CG50, 0, 1, 2.5 seconds = 1.025760511
Casio fx-991EX, .0001, .9999, 16 seconds = 1.025521692
HP 35s, 0, 1, (FIX 4) 7 seconds = 1.02574948999
HP 35s, 0, 1, (FIX 5) 14 seconds = 1.02575840691
HP 35s, 0, 1, (FIX 6) 45 seconds = 1.02576041372
HP-48G, 0, 1 4.5, (fix 5) 4.5 seconds = 1.02575840691
HP-48G, 0, 1 4.5, (fix 7) 29 seconds = 1.02576049152
TI-89, 0, 1, 18 seconds = 1.0257605126