Section 4.5 – Integration by Substitution

1. Section 4.5 – Integration by Substitution

2. The Chain Rule and Integration du u Find if . THUS:

3. Integration by Substitution Let f, g, and u be differentiable functions of x such that Then where G is an antiderivative of g.

4. Substitution Guidelines • 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. • Compute du = g '(x)dx. • Rewrite the integral in terms of u. Make sure every x and dx is no longer in the integral. • Find the resulting integral in terms of u. • Replace uby g(x)to obtain an antiderivative in terms of x. • If you have a definite integral, make sure you change the limits of integration to be in terms of ubefore you integrate.

5. Example 1 Define u and du: Substitute to replace EVERY x and dx: Integrate. Substitute back to Leave your answer in terms of x. Find .

6. Example 2 Define u and du: Substitute to replace EVERY x and dx: Solve for dx Integrate. Substitute back to Leave your answer in terms of x. Find .

7. Example 3 Rewrite Define u and du: Substitute to replace EVERY x and dx: Solve for dx Integrate.` Substitute back to Leave your answer in terms of x. Find

8. Example 4 Define u and du: Solve for dx Substitute to replace EVERY x and dx: There is still an x. Solve the initial equation of u for x. Integrate. Substitute back to Leave your answer in terms of x. Find .

9. Example 5 Define u and du: Substitute to replace EVERY x and dx: Solve for dx Find C: Find F(x) if F(0.5) = 4 and .

10. Example 6 Change the Limits: Define u and du: Substitute: Path 1: Path 2: Evaluate

11. Example 7 Change the Limits: Define u and du: Substitute: Path 1: Path 2: Evaluate

12. White Board Challenge Evaluate:

13. Integration of Even and Odd Functions Let f be integrable on the interval [-a,a]. • If f is an even function, then -a a

14. Integration of Even and Odd Functions Let f be integrable on the interval [-a,a]. • If f is an odd function, then -a a