Integration by Substitution Undoing the Chain Rule TS: Making Decisions After Reflection & Review.

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Presentation transcript:

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? Integration is a process or operation that reverses differentiation. The operation of integration determines the original function when given its derivative.

Warm Up 3 problems:

Warm Up

1 2 1 x

You have already found a function whose derivative is the expression in the integrand, so you already have an antiderivative.

Warm Up 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 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.

Integration by Substitution Take the derivative of u. Substitute into the integral. Always express your answer in terms of the original variable.

Integration by Substitution Take the derivative of u. Substitute into the integral. Always express your answer in terms of the original variable.

Integration by Substitution Take the derivative of u. Substitute into the integral. Always express your answer in terms of the original variable.

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 Substitution Always express your answer in terms of the original variable.

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.