5.5. OTHER OPTIONS FOR FINDING ALGEBRAIC ANTIDERIVATIVES
317
(no result found in terms of standard mathematical functions)
But in fact this integral is not that difficult to evaluate. If we let u = xe x , then du =
(1 + x)e x dx, which means that the preceding integral has form
(1 + x)e
x
√
1 + x 2 e 2x dx =
√
1 + u 2 du,
which is a straightforward one for any CAS to evaluate.
So, the above observations regarding computer algebra systems lead us to proceed
with some caution: while any CAS is capable of evaluating a wide range of integrals (both
definite and indefinite), there are times when the result can mislead us. We must think
carefully about the meaning of the output, whether it is consistent with what we expect,
and whether or not it makes sense to proceed.
Summary
In this section, we encountered the following important ideas:
• The method of partial fractions enables any rational function to be antidifferentiated,
because any polynomial function can be factored into a product of linear and irreducible
quadratic terms. This allows any rational function to be written as the sum of a
polynomial plus rational terms of the form
A
(x−c) n (where n is a natural number) and
Bx+C
x 2 +k
(where k is a positive real number).
• Until the development of computing algebra systems, integral tables enabled students of
calculus to more easily evaluate integrals such as
√
a 2 + u 2 du, where a is a positive
real number. A short table of integrals may be found in Appendix A.
• Computer algebra systems can play an important role in finding antiderivatives, though
we must be cautious to use correct input, to watch for unusual or unfamiliar advanced
functions that the CAS may cite in its result, and to consider the possibility that a CAS
may need further assistance or insight from us in order to answer a particular question.
Exercises
1. For each of the following integrals involving rational functions, (1) use a CAS to find the
partial fraction decomposition of the integrand; (2) evaluate the integral of the resulting
function without the assistance of technology; (3) use a CAS to evaluate the original
integral to test and compare your result in (2).
(a)
x 3 + x + 1
x 4 − 1
dx
(b)
x 5 + x 2 + 3
x 3 − 6x 2 + 11x − 6
dx
317
(no result found in terms of standard mathematical functions)
But in fact this integral is not that difficult to evaluate. If we let u = xe x , then du =
(1 + x)e x dx, which means that the preceding integral has form
(1 + x)e
x
√
1 + x 2 e 2x dx =
√
1 + u 2 du,
which is a straightforward one for any CAS to evaluate.
So, the above observations regarding computer algebra systems lead us to proceed
with some caution: while any CAS is capable of evaluating a wide range of integrals (both
definite and indefinite), there are times when the result can mislead us. We must think
carefully about the meaning of the output, whether it is consistent with what we expect,
and whether or not it makes sense to proceed.
Summary
In this section, we encountered the following important ideas:
• The method of partial fractions enables any rational function to be antidifferentiated,
because any polynomial function can be factored into a product of linear and irreducible
quadratic terms. This allows any rational function to be written as the sum of a
polynomial plus rational terms of the form
A
(x−c) n (where n is a natural number) and
Bx+C
x 2 +k
(where k is a positive real number).
• Until the development of computing algebra systems, integral tables enabled students of
calculus to more easily evaluate integrals such as
√
a 2 + u 2 du, where a is a positive
real number. A short table of integrals may be found in Appendix A.
• Computer algebra systems can play an important role in finding antiderivatives, though
we must be cautious to use correct input, to watch for unusual or unfamiliar advanced
functions that the CAS may cite in its result, and to consider the possibility that a CAS
may need further assistance or insight from us in order to answer a particular question.
Exercises
1. For each of the following integrals involving rational functions, (1) use a CAS to find the
partial fraction decomposition of the integrand; (2) evaluate the integral of the resulting
function without the assistance of technology; (3) use a CAS to evaluate the original
integral to test and compare your result in (2).
(a)
x 3 + x + 1
x 4 − 1
dx
(b)
x 5 + x 2 + 3
x 3 − 6x 2 + 11x − 6
dx