1. Integration by Substitution Antidifferentiation of a Composite Function

2. Let g be a function whose range is an interval I, and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I, then

3. Exploration Recognizing Patterns. Discover the rule using the exploration

4. Recognizing the f’(g(x)) g’(x) Pattern Evaluate

5. Recognizing the f’(g(x)) g’(x) Pattern Evaluate

6. Multiplying and Dividing by a Constant Evaluate

7. Evaluating an Integral We presently have no division rule for integrals. As a matter of fact there will never be a rule that involves division. Suggestions?

8. Change of Variables We are allowed to multiply any integral by a constant, but we can never multiply by a variable. What can we do if there is an extra variable in the given equation?

9. Integration by Substitution Evaluate There is an extra x in this integrand. In order to solve this equation, we will let u = 2x – 1.

10. Integration by Substitution Now solve this equation for x

11. Integration by Substitution

12. This is a correct answer, but it can be simplified a little further. If the problem is not a multiple choice question on the test or quiz, this answer is completed. If it is a multiple choice question, it will be given as

13. Integration by Substitution

14. Integrating Trig Functions Remember that you need the derivative of the trig function and the function.

15. Guidelines for Making a Change of Variable 1. Choose a substitution u = g(x). Usually, it is best to choose the inner part of a composite function, such as a quantity raised to a power. 2.Compute du = g’(x) dx. 3.Rewrite the integral in terms of the variable u. 4.Evaluate the resulting integral in terms of u. 5.Replace u by g(x) to obtain an antiderivative in terms of x. 6.Check your answer by differentiating.

16. General Power Rule for Integration If g is a differentiable function of x, then

17. Definite Integrals

18. Integration of Even and Odd Functions Let f be integrable on the closed interval, [a, –a ], 1. If f is an even function, then 2. If f is an odd function, then