in this section, we introduce the idea of the indefinite integral. we also look at the process of...
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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
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Definition
For any function f, is called the indefinite
integral of f and represents the most general
antiderivative of f.
![Page 3: 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](https://reader035.vdocument.in/reader035/viewer/2022062222/5697bf881a28abf838c8960b/html5/thumbnails/3.jpg)
Definition
For any function f, is called the indefinite
integral of f and represents the most general
antiderivative of f.
For example:
![Page 4: 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](https://reader035.vdocument.in/reader035/viewer/2022062222/5697bf881a28abf838c8960b/html5/thumbnails/4.jpg)
Example 1
Find each of the following:
(a)
(b)
(c)
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Example 1cont.
Find each of the following:
(d)
(e)
(f)
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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.
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TheoremChange of Variables in an
Integral
Let f, u, and g be continuous functions such that:
for all x.
Then:
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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
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Our Example
Looking again at
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Example 2
Find each of the following:
(a)
(b)
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Example 3
Find each of the following:
(a)
(b)
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Example 4
Find each of the following:
(a)
(b)
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TheoremChange of Variables in a Definite
Integral
Let f, u, and g be continuous functions such that:
for all x.
Then:
![Page 14: 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](https://reader035.vdocument.in/reader035/viewer/2022062222/5697bf881a28abf838c8960b/html5/thumbnails/14.jpg)
Example 5
Find each of the following:
(a)
(b)