5.5. OTHER OPTIONS FOR FINDING ALGEBRAIC ANTIDERIVATIVES
315
For the purposes of this text, when we need to use a CAS, we are going to turn instead
to a similar, but somewhat different computational tool, the web-based “computational
knowledge engine” called WolframAlpha. There are two features of WolframAlpha that
make it stand out from the CAS options mentioned above: (1) unlike Maple and Mathematica, WolframAlpha is free (provided we are willing to suffer through some pop-up
advertising); and (2) unlike any of the three, the syntax in WolframAlpha is flexible. Think
of WolframAlpha as being a little bit like doing a Google search: the program will interpret
what is input, and then provide a summary of options.
If we want to have WolframAlpha evaluate an integral for us, we can provide it syntax
such as
integrate xˆ2 dx
to which the program responds with
x
2 dx =
x 3
3
+ constant.
While there is much to be enthusiastic about regarding CAS programs such as WolframAlpha, there are several things we should be cautious about: (1) a CAS only responds to
exactly what is input; (2) a CAS can answer using powerful functions from highly advanced
mathematics; and (3) there are problems that even a CAS cannot do without additional
human insight.
Although (1) likely goes without saying, we have to be careful with our input: if we
enter syntax that defines a function other than the problem of interest, the CAS will work
with precisely the function we define. For example, if we are interested in evaluating the
integral
1
16 − 5x 2 dx,
and we mistakenly enter
integrate 1/16 - 5xˆ2 dx
a CAS will (correctly) reply with
1
16
x −
5
3
x
3 .
It is essential that we are sufficiently well-versed in antidifferentiation to recognize that this
function cannot be the one that we seek: integrating a rational function such as
1
16−5x 2 ,
we expect the logarithm function to be present in the result.
Regarding (2), even for a relatively simple integral such as
1
16−5x 2 dx, some CASs
will invoke advanced functions rather than simple ones. For instance, if we use Maple to
execute the command
315
For the purposes of this text, when we need to use a CAS, we are going to turn instead
to a similar, but somewhat different computational tool, the web-based “computational
knowledge engine” called WolframAlpha. There are two features of WolframAlpha that
make it stand out from the CAS options mentioned above: (1) unlike Maple and Mathematica, WolframAlpha is free (provided we are willing to suffer through some pop-up
advertising); and (2) unlike any of the three, the syntax in WolframAlpha is flexible. Think
of WolframAlpha as being a little bit like doing a Google search: the program will interpret
what is input, and then provide a summary of options.
If we want to have WolframAlpha evaluate an integral for us, we can provide it syntax
such as
integrate xˆ2 dx
to which the program responds with
x
2 dx =
x 3
3
+ constant.
While there is much to be enthusiastic about regarding CAS programs such as WolframAlpha, there are several things we should be cautious about: (1) a CAS only responds to
exactly what is input; (2) a CAS can answer using powerful functions from highly advanced
mathematics; and (3) there are problems that even a CAS cannot do without additional
human insight.
Although (1) likely goes without saying, we have to be careful with our input: if we
enter syntax that defines a function other than the problem of interest, the CAS will work
with precisely the function we define. For example, if we are interested in evaluating the
integral
1
16 − 5x 2 dx,
and we mistakenly enter
integrate 1/16 - 5xˆ2 dx
a CAS will (correctly) reply with
1
16
x −
5
3
x
3 .
It is essential that we are sufficiently well-versed in antidifferentiation to recognize that this
function cannot be the one that we seek: integrating a rational function such as
1
16−5x 2 ,
we expect the logarithm function to be present in the result.
Regarding (2), even for a relatively simple integral such as
1
16−5x 2 dx, some CASs
will invoke advanced functions rather than simple ones. For instance, if we use Maple to
execute the command