5.3 integration by substitution dfs-102
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Calculus 6.2
Calculus-IIMath-102Dr. Farhana Shaheen5.3 Integration by substitution
Always Remember:Integration is by Observation And The Best way is by Substitution
Integrals of Transcendental FunctionsTrigonometric functionsThe inverse trigonometric functionsExponential functionsLogarithms.
INTEGRATION BY SUBSTITUTION
Note: Integration by substitution can be used for a variety of integrals: some compound functions, some products and some quotients.
Sometimes we have a choice of method.
Choosing u
Guidelines for u-Substitution (p. 334)
(Page: 337)Equation (5)
Equation (6) Equation (7)
The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for only a limited range of functions? We can sometimes use substitution or change of variable to rewrite functions in a form that we can integrate.
Example 1:
The variable of integration must match the variable in the expression.
Dont forget to substitute the value for u back into the problem!
Example 2:
One of the clues that we look for is if we can find a function and its derivative in the integral.
The derivative of is .
Note that this only worked because of the 2x in the original.Many integrals can not be done by substitution.
Example 3:
Solve for dx.
Example 4:
Example 5:
We solve for because we can find it in the integral.
Example 6:
Example: 7
Example: 8
Example: 10 (page 336) Evaluate:
Example: 11 (page 336) Evaluate:
Exercises
Use substitution to integrate the following. (Where possible, you could also use a 2nd method.)
1.
2.
3.
Question for you:
Ex: 5.3 Questions: 1 54 (page 338)
Questions: 1 12 (substitution is given)Questions: 15 54 (using substitution)
Questions: 61 62 (using substitution and formulas)
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