integration by substitution undoing the chain rule ts: making decisions after reflection &...
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Integration by SubstitutionIntegration by SubstitutionUndoing the Chain RuleUndoing the Chain Rule
TS: Making Decisions After Reflection TS: Making Decisions After Reflection & Review& Review
ObjectiveObjective
To evaluate integrals using the technique To evaluate integrals using the technique of integration by substitution.of integration by substitution.
Warm UpWarm UpWhat is a synonym for the term integration?
Antidifferentation
What is integration?
Integration is a process or operation that reverses differentiation.Integration is a process or operation that reverses differentiation.
The operation of integration determines the original function when given its derivative.
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43Find if 5 2dydx y x x
2Find if ln 3dydx y x
32 3Evaluate 4 15 +2 5 2 x x x dx
3 problems:
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43Find if 5 2dydx y x x
dydx 34 5 2x x
3 215 2x
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2Find if ln 3dydx y x
dydx 2
13
x
61 x
1
2
2dydx x
1
x
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32 3Evaluate 4 15 +2 5 2 x x x dx
You have already found a function whose derivative is the expression in the integrand, so you already have an antiderivative.
43Find if 5 2dydx y x x
dydx 34 5 2x x
3 215 2x
435 2x x C
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32 3Evaluate 4 15 +2 5 2 x x x dx
435 2x x C
Inside functionDerivative of inside function
Some integrals are chain rule problems in reverse.
If the derivative of the inside function is sitting elsewhere in the integrand, then you can use a technique called
integration by substitution to evaluate the integral.
Integration by SubstitutionIntegration by Substitution
One method for evaluating integrals involves One method for evaluating integrals involves untangling the chain rule.untangling the chain rule.
This technique is called integration by This technique is called integration by substitution.substitution.
Integration by substitution is a technique for Integration by substitution is a technique for finding the antiderivative of a composite finding the antiderivative of a composite function.function.
Integration by SubstitutionIntegration by Substitution
4Evaluate 3 3 4 x dx
Let 3 4u x
43 u dx
3dudx dx dx
3 du dx
4 3 u dx 4 u du
5
5u C
515 u C
515 3 4x C
Take the derivative of u.
Substitute intothe integral.
Always express your answer in terms of the original variable.
Integration by SubstitutionIntegration by Substitution
202Evaluate 42 +4 x x dx2Let 4u x
2042 x u dx
2dudx xdx dx
2 du x dx
2042 u x dx
20 1242 u du
12 du x dx
2021 u du
212121u C 21u C
212 4x C
Take the derivative of u.
Substitute intothe integral.
Always express your answer in terms of the original variable.
Integration by SubstitutionIntegration by Substitution2 3Evaluate 3 5 x x dx
3Let 5u x
1
22 33 5 x x dx
23dudx xdx dx
23 du x dx
1
223 x u dx1
2 23 u x dx 1
2 u du 3
2
32
u C 3
223 u C
3
232
3 5 x C
Take the derivative of u.
Substitute intothe integral.
Always express your answer in terms of the original variable.
Integration by SubstitutionIntegration by Substitution
Experiment with different choices for Experiment with different choices for uu when using integration by substitution.when using integration by substitution.
A good choice is one whose derivative is A good choice is one whose derivative is expressed elsewhere in the integrand.expressed elsewhere in the integrand.
Integration by SubstitutionIntegration by Substitution
32Evaluate 2 1 x x x dx 2Let u x x
32 1 x u dx
2 1dudx x dx dx
2 1 du x dx
32 1 u x dx
3 u du 4
4u C
414 u C
4214 x x C
Always express your answer in terms of the original variable.
ConclusionConclusion
To integrate by substitution, select an To integrate by substitution, select an expression for expression for uu. .
A good choice for A good choice for uu is one whose is one whose derivative is expressed elsewhere in the derivative is expressed elsewhere in the integrand.integrand.
Next, rewrite the integral in terms of Next, rewrite the integral in terms of uu. . Then, simplify the integral and evaluate.Then, simplify the integral and evaluate.