5.4. INTEGRATION BY PARTS
299
5.4 Integration by Parts
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How do we evaluate indefinite integrals that involve products of basic functions
such as
x sin(x) dx and
xe x dx?
• What is the method of integration by parts and how can we consistently apply it
to integrate products of basic functions?
• How does the algebraic structure of functions guide us in identifying u and dv in
using integration by parts?
Introduction
In Section 5.3, we learned the technique of u-substitution for evaluating indefinite integrals that involve certain composite functions. For example, the indefinite integral
x 3 sin(x 4 ) dx is perfectly suited to u-substitution, since not only is there a composite
function present, but also the inner function’s derivative (up to a constant) is multiplying
the composite function. Through u-substitution, we learned a general situation where
recognizing the algebraic structure of a function can enable us to find its antiderivative.
It is natural to ask similar questions to those we considered in Section 5.3 about
functions with a different elementary algebraic structure: those that are the product of
basic functions. For instance, suppose we are interested in evaluating the indefinite integral
x sin(x) dx.
Here, there is not a composite function present, but rather a product of the basic functions
f (x) = x and g(x) = sin(x). From our work in Section 2.3 with the Product Rule, we know
that it is relatively complicated to compute the derivative of the product of two functions,
so we should expect that antidifferentiating a product should be similarly involved. In
addition, intuitively we expect that evaluating
x sin(x) dx will involve somehow reversing
the Product Rule.
To that end, in Preview Activity 5.4 we refresh our understanding of the Product Rule
and then investigate some indefinite integrals that involve products of basic functions.
Preview Activity 5.4. In Section 2.3, we developed the Product Rule and studied how it
is employed to differentiate a product of two functions. In particular, recall that if f and g
299
5.4 Integration by Parts
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How do we evaluate indefinite integrals that involve products of basic functions
such as
x sin(x) dx and
xe x dx?
• What is the method of integration by parts and how can we consistently apply it
to integrate products of basic functions?
• How does the algebraic structure of functions guide us in identifying u and dv in
using integration by parts?
Introduction
In Section 5.3, we learned the technique of u-substitution for evaluating indefinite integrals that involve certain composite functions. For example, the indefinite integral
x 3 sin(x 4 ) dx is perfectly suited to u-substitution, since not only is there a composite
function present, but also the inner function’s derivative (up to a constant) is multiplying
the composite function. Through u-substitution, we learned a general situation where
recognizing the algebraic structure of a function can enable us to find its antiderivative.
It is natural to ask similar questions to those we considered in Section 5.3 about
functions with a different elementary algebraic structure: those that are the product of
basic functions. For instance, suppose we are interested in evaluating the indefinite integral
x sin(x) dx.
Here, there is not a composite function present, but rather a product of the basic functions
f (x) = x and g(x) = sin(x). From our work in Section 2.3 with the Product Rule, we know
that it is relatively complicated to compute the derivative of the product of two functions,
so we should expect that antidifferentiating a product should be similarly involved. In
addition, intuitively we expect that evaluating
x sin(x) dx will involve somehow reversing
the Product Rule.
To that end, in Preview Activity 5.4 we refresh our understanding of the Product Rule
and then investigate some indefinite integrals that involve products of basic functions.
Preview Activity 5.4. In Section 2.3, we developed the Product Rule and studied how it
is employed to differentiate a product of two functions. In particular, recall that if f and g