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Lecture 8 (Integrals continued)
Indefinite Integrals
Recall: Part I of FTC says that if is continuous, then ∫ ( )
is an
antiderivative of . Part II tells us that ∫ ( )
( ) ( ), where is an
antiderivative. From now on we shall use the following notation for an
antiderivative:
∫ ( ) ( )
and call it an indefinite integral.
Example:
∫
∫
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Caution: There is a difference between a definite integral ∫ ( )
which is a number, and an indefinite integral ∫ ( ) which is a function (or a
family of functions).
∫ ( )
∫ ( ) |
Properties of indefinite integrals:
∫[ ( ) ( )] ∫ ( ) ∫ ( )
∫ ( ) ∫ ( )
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Table of Indefinite Integrals
∫
∫
∫
∫ ∫
∫ ∫
∫ ∫
∫ ∫
∫
∫
√
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Examples: Find the indefinite integrals
∫
∫ √
∫( )
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Examples: Evaluate
∫( )
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∫
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∫
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Substitution Rule
Consider the following integral:
∫
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In general, this method will work if we have an integral of the form
∫ ( ( )) ( )
If is an antiderivative of , then . Thus,
∫ ( ( )) ( ) ( ( )) (by the Chain Rule)
If we make a substitution ( ), then ( ) , and we get that
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Substitution Rule: If ( ) is a differentiable function whose range is an
interval on which is continuous, then
∫ ( ( )) ( ) ∫ ( )
Example: Evaluate
∫√
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∫
∫ ( )
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∫
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∫
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∫ √
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∫
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Substitution Rule for Definite Integrals:
If is continuous on [ ] and is continuous on the range of ( ), then
∫ ( ( )) ( )
∫ ( ) ( )
( )
Proof:
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Example: Evaluate
∫ √
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∫
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∫
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∫ ( )
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Symmetry
Suppose is continuous on [ ].
If is even, i.e. ( ) ( ), then ∫ ( )
– ∫ ( )
If is odd, i.e. ( ) ( ), then ∫ ( )
–
Proof:
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Proof (continued):
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Example: Evaluate
∫