1. Integration Techniques: Substitution
OBJECTIVE
Evaluate integrals using substitution.
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2. 4.5 Integration Techniques: Substitution
The following formulas provide a basis for an
integration technique called substitution.
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3. 4.5 Integration Techniques: Substitution
Example 1: For y = f (x) = x3, find dy.
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4. 4.5 Integration Techniques: Substitution
Example 2: For u = F(x) = x2/3, find du.
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5. 4.5 Integration Techniques: Substitution
Example 3: For find dy.
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6. 4.5 Integration Techniques: Substitution
Quick Check 1
Find each differential.
a.)
b.)
c.)
d.)
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7. 4.5 Integration Techniques: Substitution
Quick Check 1 Continued
a.)
b.)
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8. 4.5 Integration Techniques: Substitution
Quick Check 1 Concluded
c.)
d.)
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9. 4.5 Integration Techniques: Substitution
Example 4: Evaluate:
Note that 3x2 is the derivative of x3. Thus,
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10. 4.5 Integration Techniques: Substitution
Quick Check 2
Evaluate:
Note that
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11. 4.5 Integration Techniques: Substitution
Example 5: Evaluate:
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12. 4.5 Integration Techniques: Substitution
Example 6: Evaluate:
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13. 4.5 Integration Techniques: Substitution
Quick Check 3
Evaluate:
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14. 4.5 Integration Techniques: Substitution
Example 7: Evaluate:
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15. 4.5 Integration Techniques: Substitution
Example 8: Evaluate:
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16. 4.5 Integration Techniques: Substitution
Quick Check 4
Evaluate:
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17. 4.5 Integration Techniques: Substitution
Example 9: Evaluate:
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18. 4.5 Integration Techniques: Substitution
Example 10: Evaluate:
We first find the indefinite integral and then evaluate
the integral over [0, 1].
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19. 4.5 Integration Techniques: Substitution
Example 10 (concluded):
Then, we have
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20. 4.5 Integration Techniques: Substitution
Section Summary
Integration by substitution is the reverse of applying the Chain Rule of Differentiation.
The substitution is reversed after the integration has been performed.
Results should be checked using differentiation.
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