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Integration by Substitution. Undoing the Chain Rule TS: Making Decisions After Reflection & Review. Objective. To evaluate integrals using the technique of integration by substitution. Warm Up. What is a synonym for the term integration?. Antidifferentation. What is integration?.

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Integration by substitution

Integration by Substitution

Undoing the Chain Rule

TS: Making Decisions After Reflection & Review


Objective
Objective

  • To evaluate integrals using the technique of integration by substitution.


Warm up
Warm Up

What is a synonym for the term integration?

Antidifferentation

What is integration?

Integration is a process or operation that reverses differentiation.

The operation of integration determines the

original function when given its derivative.


Warm up1
Warm Up

3 problems:



Warm up3
Warm Up

2

1

x

1


Warm up4
Warm Up

You have already found a function whose derivative is the expression

in the integrand, so you already have an antiderivative.


Warm up5
Warm Up

Derivative of

inside function

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 substitution1
Integration by Substitution

  • One method for evaluating integrals involves untangling the chain rule.

  • This technique is called integration by substitution.

  • Integration by substitution is a technique for finding the antiderivative of a composite function.

CHAIN RULE


Integration by substitution2
Integration by Substitution

Take the

derivative of u.

Substitute into

the integral.

Always express your

answer in terms of the

original variable.


Integration by substitution3
Integration by Substitution

Take the

derivative of u.

Substitute into

the integral.

Always express your

answer in terms of the

original variable.


Integration by substitution4
Integration by Substitution

Take the

derivative of u.

Substitute into

the integral.

Always express your

answer in terms of the

original variable.


Integration by substitution5
Integration by Substitution

  • Experiment with different choices for u when using integration by substitution.

  • A good choice is one whose derivative is expressed elsewhere in the integrand.


Integration by substitution6
Integration by Substitution

Always express your

answer in terms of the

original variable.


Conclusion
Conclusion

  • To integrate by substitution, select an expression for u.

  • A good choice for u is one whose derivative is expressed elsewhere in the integrand.

  • Next, rewrite the integral in terms of u.

  • Then, simplify the integral and evaluate.