HP50 : indefinite integral of '0^x' gives '?' and 0 for '0*x'

[Gil]

Try on the HP50G
'0^X' INTVX (integral of 0^X).

But S (0^X) dX
is equivalent to S (0) dX = Constant,
as the derivative of a constant is equal to 0.

But the HP50G returns '?'.

Besides, if you write directly 0 INTVX (integral of 0),
the calculator returns 0 (instead of Constant).

Somewhat disappointing.

However, the HP50G gives correctly the numerical area (with the command —>NUM, but not with EVAL command) from a to b for the function '0^X': 0, as [b-a] × 0 = 0.

Does another calculator like the Prime, or CASIO/TI gives the expected answers?

[rawi]

This seems to be a strange problem for calculators:
HP Prime: G1 and G2:
Integral 0^x without lower and upper limit in CAS: undef.
Integral from a to b of 0^x --> invalid input (ungültige Eingabe, home) or undef. (CAS)
Integral from 1 to 2 of 0^x --> invalid input (Home) or undef. (CAS)

TI 89 Titanium and TI nspire CX II T CAS:
Integral 0^x without lower and upper limit: no result (input is repeated as result)
Integral from a to b of 0^x --> no result (input is repeated)
Integral from 1 to 2 of 0^x --> 0

I think the reason why the integrals from a to b do not work is that 0^0 is undefined in all the calculators (there seems to be a new definition among mathematicicians but I am not sure about that). But the integral from 1 to 2 should be no problem.

[rawi]

Some additional information:
Casio CP 400: Integral from 1 to 2 of 0^x: Undefined.

HP 15C:
If you integrate the follwing program:

LBL A
0
X<>Y
Y^X
g RTN

from 0 to 1 or from 1 to 2 the result is always 0.

As well the TI 89 Titanium and the TI nsprire get the integral from 0 to 1 right with the result 0.

[klesl]

According to me the function 0^x is undefined
khicas + casio fx-cg50 - undefined
classpad II - undefined
xcas - undefined
ti-nspire cx cas - no result
mathcad 15 - no result
matlab 2023- no result
no result means "input is repeated as result"

int(a^x)=a^x/ln(a)
for a=0 the result is not defined
and 0^0 is indeterminate form

[Gil]

About the last answer
int(a^x)=a^x/ln(a)

to be better formulated, with x>0:
int(a^x)=a^x/ln(a) for a≠0
int(a^x)=constant, else (for a=0) ?

Regarding that "stupid" function 0^X, it seems to be defined for X>0.

Note that the area (zero!) is correct
when setting real bounds and then the numerical calculation is requested (—>NUM) with the HP50G, but not with the EVAL command.

But if you put bounds like variables named a and b, you won't get the expected 0 value with the HP50G, supposing of course x>0 (or the infinity when x<0).

[klesl]

small corerction: int(a^x)=a^x/ln(a) for a>0
There is no way how to compute int(a^x,x) for a=0 because you need to rewrite a^x as e^(x*ln(a)) and logarithm is defined for a>0 only. This integral isn't equal to constant, it is undef as logarithm.
Also you don't find int(a^x,x) for a=0 in any math tables.

[Gil]

Ok for the undefined function '0^x'.

But what about the well defined function '0*x'?

Integral of (0*x) should be simplified before calculating and (logically?) be equal to integral of (0) = constant (as correctly answers Wolfram Alpha for the aforementioned indefinite integral).

Or am I wrong?

On the HP50G, however, '0*X' INTVX (with no bounds) gives zero, as it has been initially calculated with a general formulae a*x²/2, and then the factor a is replaced by its original value a=0, which gives a final value of 0, a special case — indeed a correct result when settling real bounds with the HP50G and executing EVAL command
or —>NUM command.