In this section, we introduce the idea of the indefinite integral. We also look at the process of variable substitution to find antiderivatives of more complex functions.
Section 5.4 Finding Antiderivatives: Substitution
Definition
For any function f, is called the indefinite
integral of f and represents the most general
antiderivative of f.
Definition
For any function f, is called the indefinite
integral of f and represents the most general
antiderivative of f.
For example:
Example 1
Find each of the following:
(a)
(b)
(c)
Example 1cont.
Find each of the following:
(d)
(e)
(f)
Substitution
What if the integrand is not something that we recognize as a “basic” antiderivative rule?
For example,
We would like to do a variable substitution so that the “new” integral is one that we recognize as a “basic” antiderivative rule.
TheoremChange of Variables in an
Integral
Let f, u, and g be continuous functions such that:
for all x.
Then:
The Process
Substitute: Choose a function u = u(x) such that the substitution of u for x and du for dx changes into
Antidifferentiate: Solve - that is, find G(u) such that
Resubstitute: Substitute x back in for u to get the answer to have an antiderivative of the original function
Our Example
Looking again at
Example 2
Find each of the following:
(a)
(b)
Example 3
Find each of the following:
(a)
(b)
Example 4
Find each of the following:
(a)
(b)
TheoremChange of Variables in a Definite
Integral
Let f, u, and g be continuous functions such that:
for all x.
Then:
Example 5
Find each of the following:
(a)
(b)
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