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5.5. OTHER OPTIONS FOR FINDING ALGEBRAIC ANTIDERIVATIVES
5.5 Other Options for Finding Algebraic Antiderivatives
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How does the method of partial fractions enable any rational function to be
antidifferentiated?
• What role have integral tables historically played in the study of calculus and how
can a table be used to evaluate integrals such as
√
a 2 + u 2 du?
• What role can a computer algebra system play in the process of finding antiderivatives?
Introduction
In the preceding sections, we have learned two very specific antidifferentiation techniques:
u-substitution and integration by parts. The former is used to reverse the chain rule,
while the latter to reverse the product rule. But we have seen that each only works
in very specialized circumstances. For example, while
xe x 2 dx may be evaluated by
u-substitution and
xe x dx by integration by parts, neither method provides a route to
evaluate
e x 2 dx. That fact is not a particular shortcoming of these two antidifferentiation
techniques, as it turns out there does not exist an elementary algebraic antiderivative for
e x 2 . Said differently, no matter what antidifferentiation methods we could develop and
learn to execute, none of them will be able to provide us with a simple formula that does
not involve integrals for a function F(x) that satisfies F ′ (x) = e x 2 .
In this section of the text, our main goals are to better understand some classes of
functions that can always be antidifferentiated, as well as to learn some options for so
doing. At the same time, we want to recognize that there are many functions for which an
algebraic formula for an antiderivative does not exist, and also appreciate the role that
computing technology can play in helping us find antiderivatives of other complicated
functions. Throughout, it is helpful to remember what we have learned so far: how to
reverse the chain rule through u-substitution, how to reverse the product rule through
integration by parts, and that overall, there are subtle and challenging issues to address
when trying to find antiderivatives.
Preview Activity 5.5. For each of the indefinite integrals below, the main question is to
decide whether the integral can be evaluated using u-substitution, integration by parts,
a combination of the two, or neither. For integrals for which your answer is affirmative,
state the substitution(s) you would use. It is not necessary to actually evaluate any of the
integrals completely, unless the integral can be evaluated immediately using a familiar
5.5. OTHER OPTIONS FOR FINDING ALGEBRAIC ANTIDERIVATIVES
5.5 Other Options for Finding Algebraic Antiderivatives
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How does the method of partial fractions enable any rational function to be
antidifferentiated?
• What role have integral tables historically played in the study of calculus and how
can a table be used to evaluate integrals such as
√
a 2 + u 2 du?
• What role can a computer algebra system play in the process of finding antiderivatives?
Introduction
In the preceding sections, we have learned two very specific antidifferentiation techniques:
u-substitution and integration by parts. The former is used to reverse the chain rule,
while the latter to reverse the product rule. But we have seen that each only works
in very specialized circumstances. For example, while
xe x 2 dx may be evaluated by
u-substitution and
xe x dx by integration by parts, neither method provides a route to
evaluate
e x 2 dx. That fact is not a particular shortcoming of these two antidifferentiation
techniques, as it turns out there does not exist an elementary algebraic antiderivative for
e x 2 . Said differently, no matter what antidifferentiation methods we could develop and
learn to execute, none of them will be able to provide us with a simple formula that does
not involve integrals for a function F(x) that satisfies F ′ (x) = e x 2 .
In this section of the text, our main goals are to better understand some classes of
functions that can always be antidifferentiated, as well as to learn some options for so
doing. At the same time, we want to recognize that there are many functions for which an
algebraic formula for an antiderivative does not exist, and also appreciate the role that
computing technology can play in helping us find antiderivatives of other complicated
functions. Throughout, it is helpful to remember what we have learned so far: how to
reverse the chain rule through u-substitution, how to reverse the product rule through
integration by parts, and that overall, there are subtle and challenging issues to address
when trying to find antiderivatives.
Preview Activity 5.5. For each of the indefinite integrals below, the main question is to
decide whether the integral can be evaluated using u-substitution, integration by parts,
a combination of the two, or neither. For integrals for which your answer is affirmative,
state the substitution(s) you would use. It is not necessary to actually evaluate any of the
integrals completely, unless the integral can be evaluated immediately using a familiar