Integration by Substitution 4.5 Day 1 On the agenda: How to Integrate using u-substitution aka Change of Variable(s) HW: p. 297 # 3-33 mx3, 41-49 odd

Role of Substitution in Integration Substitution is a technique that we can sometimes use to rewrite functions in a form that we can integrate. Recognition is KEY! Two Components to keep in mind: Look for a derivative one term to be somewhere in the integral. Do not worry about coefficients since we can ‘adjust’ for them

Don’t forget to substitute the value for u back into the problem! Example 1 Don’t forget to substitute the value for u back into the problem! 11/21/2018

The variable of integration must match the variable in the expression. Example 2: The variable of integration must match the variable in the expression. Don’t forget to substitute the value for u back into the problem! 11/21/2018

Note that this only worked because of the 2x in the original. Example 3: The derivative of is . Note that this only worked because of the 2x in the original. Many integrals can not be done by substitution. 11/21/2018

You Try It First Example 4: 11/21/2018

Here U-Sub does not work Example 5 Here U-Sub does not work 11/21/2018

You Try It First Example 6: 11/21/2018

You Try It First Example 7: 11/21/2018

You Try It First Example 8: 11/21/2018

Change of Variables Wrong! The limits don’t match! Example 5: Using the original limits: Leave the limits out until you substitute back. Wrong! The limits don’t match! This is usually more work than finding new limits 11/21/2018

We can find new limits, and then we don’t have to substitute back. Example 5: new limit We can find new limits, and then we don’t have to substitute back. new limit We could have substituted back and used the original limits. 11/21/2018

Don’t forget to use the new limits. Example 6: Don’t forget to use the new limits. 11/21/2018