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# Integration by Substitution - PowerPoint PPT Presentation

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

Undoing the Chain Rule

TS: Making Decisions After Reflection & Review

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

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.

3 problems:

2

1

x

1

You have already found a function whose derivative is the expression

in the integrand, so you already have an antiderivative.

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.

• 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

Take the

derivative of u.

Substitute into

the integral.

Always express your

answer in terms of the

original variable.

Take the

derivative of u.

Substitute into

the integral.

Always express your

answer in terms of the

original variable.

Take the

derivative of u.

Substitute into

the integral.

Always express your

answer in terms of the

original variable.

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

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

Always express your

answer in terms of the

original variable.

• 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.