Strange Integration "bug"

[Wes Loewer]

The Prime CAS has some very powerful features which sometimes produces results that might be different from what you expected, but usually for good reasons.

In this case, the sqrt(1-x^2)(x+2) does not mean what you probably think it means.
For instance, start with:
f(x):=x^2+1
g(x):=x^3

The following is a valid syntax used in textbooks, but not usually allowed on calculators but supported on the Prime:
(f*g)(2) --> 40
(f+g)(x+1) --> (x+1)^3+(x+1)^2+1

Notice how the adjacent parentheses do not mean implied multiplication here. They mean "apply the function f+g to x+1". If you use your expression sqrt(1-x^2)(x+2) without the integral, you get:
sqrt(1-x^2)(x+2) --> √(1-(x+2)^2)

which is correct if you interpret the sqrt(1-x^2) as a function which is using (x+2) as the argument.

In general I tell my students to avoid using implied multiplication on any CAS except for the simplest of cases, like 2x+5y=6. The Nspire CAS also has cases that confuse my students, like x(x+1) being rejected since it could be implied multiplication or it could be a function named x. The inspire allows y(x+1) or x(y+1) as functions. As humans, we would likely interpret y(x+1) as a function, but we might interpret x(y+1) as implied multiplication. Understanding human thought is tricky business. :-)

Even simple things like 1/2pi has different meaning on different calculators, even different models from the same brand.