From the definition of indefinite integral of
we know
if and only if
is an antiderivative of
Therefore, when claiming that
it is important to check whether this statement is correct by verifying that
Verifying an indefinite integral
Each of the following statements is of the form
Verify that each statement is correct by showing that
Since
the statement
is correct.
Note that we are verifying an indefinite integral for a sum. Furthermore,
and
are antiderivatives of
and
respectively, and the sum of the antiderivatives is an antiderivative of the sum. We discuss this fact again later in this section.
Using the product rule, we see that
Therefore, the statement
is correct.
Note that we are verifying an indefinite integral for a product. The antiderivative
is not a product of the antiderivatives. Furthermore, the product of antiderivatives,
is not an antiderivative of
since
In general, the product of antiderivatives is not an antiderivative of a product.
In
[link] , we listed the indefinite integrals for many elementary functions. Let’s now turn our attention to evaluating indefinite integrals for more complicated functions. For example, consider finding an antiderivative of a sum
In
[link] a. we showed that an antiderivative of the sum
is given by the sum
—that is, an antiderivative of a sum is given by a sum of antiderivatives. This result was not specific to this example. In general, if
and
are antiderivatives of any functions
and
respectively, then
Therefore,
is an antiderivative of
and we have
Similarly,
In addition, consider the task of finding an antiderivative of
where
is any real number. Since
for any real number
we conclude that
These properties are summarized next.
Properties of indefinite integrals
Let
and
be antiderivatives of
and
respectively, and let
be any real number.
Sums and Differences
Constant Multiples
From this theorem, we can evaluate any integral involving a sum, difference, or constant multiple of functions with antiderivatives that are known. Evaluating integrals involving products, quotients, or compositions is more complicated (see
[link] b. for an example involving an antiderivative of a product.) We look at and address integrals involving these more complicated functions in
Introduction to Integration . In the next example, we examine how to use this theorem to calculate the indefinite integrals of several functions.
Questions & Answers
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1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
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Kamara
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