Integrals involving NORMALD_CDF - possible CAS bug?

[Nigel (UK)]

I tried to evaluate the integral \[\int_{-10}^{10} \hbox{NORMALD}(X)*\hbox{NORMALD_CDF}(X)\,{\rm d}X \] in the Home screen, expecting to get an answer close to \(0.5\). Instead I got an error message - "Error: Undefined result". This was strange - the integrand is perfectly well-behaved and gives a smooth graph in the Function app.

Switching to CAS mode cast some light on the situation. Copying and pasting the above integral to the CAS screen, with Exact mode selected, gave a different error message: "Error while checking exact value with approximate value - returning both!" The calculator then returned the following list: \[\left[{1\over2}*{\rm erf}(5*\sqrt{2})-{1\over2}*{\rm erf}(-5*\sqrt{2})\qquad 0.5\right].\] Evaluation of the exact expression returns 1, which is very different to 0.5! Assuming that the Home screen asks the CAS to help with the integral, it is not surprising that it is unable to return a result.

The exact expression in the above list is incorrect. It is the value of the integral without the NORMAL_CDF function. This is, presumably, a CAS bug.

An easy work-around is to turn off exact evaluation in the CAS settings. All errors then vanish.

Curiously, entering \[\int_{-10}^{10} \hbox{NORMALD}(X)*\hbox{normald_cdf}(X)\,{\rm d}X \] in CAS mode (note the lower-case letters!) returns an unevaluated but correctly expanded integral in exact mode. Strange!

All of this works in the same way both on my actual G1 calculator and on the Windows Prime. Both are running firmware 20200116.

Nigel (UK)