316
5.5. OTHER OPTIONS FOR FINDING ALGEBRAIC ANTIDERIVATIVES
int(1/(16-5*xˆ2), x);
the program responds with
1
16 − 5x 2 dx =
√
5
20
arctanh(
√
5
4
x).
While this is correct (save for the missing arbitrary constant, which Maple never reports),
the inverse hyperbolic tangent function is not a common nor familiar one; a simpler way
to express this function can be found by using the partial fractions method, and happens
to be the result reported by WolframAlpha:
1
16 − 5x 2 dx =
1
8
√
5
log(4
√
5 + 5
√
x) − log(4
√
5 − 5
√
x)
+ constant.
Using sophisticated functions from more advanced mathematics is sometimes the way
a CAS says to the user “I don’t know how to do this problem.” For example, if we want to
evaluate
e
−x 2 dx,
and we ask WolframAlpha to do so, the input
integrate exp(-xˆ2) dx
results in the output
e
−x 2 dx =
√
π
2
erf(x) + constant.
The function “erf(x)” is the error function, which is actually defined by an integral:
erf(x) =
2
√
π
x
0
e
−t 2 dt.
So, in producing output involving an integral, the CAS has basically reported back to us
the very question we asked.
Finally, as remarked at (3) above, there are times that a CAS will actually fail without
some additional human insight. If we consider the integral
(1 + x)e
x
√
1 + x 2 e 2x dx
and ask WolframAlpha to evaluate
int (1+x) * exp(x) * sqrt(1+xˆ2 * exp(2x)) dx,
the program thinks for a moment and then reports