1. Math 155 – Calculus 2 Andy Rosen www.rosenmath.com PLEASE SIGN IN

2. Substitution Right now, we can only integrate a small number of functions…in this chapter we will learn two methods that will expand that list. Ex. If f (x) = (x 10 – 4) 4, find f (x).

3. Ex.

4. Substitution is a result of the chain rule u will be the stuff on the inside its derivative will always show up on the outside  All x’s must change to u’s before we can integrate

5. Ex.

7. Pract. 1. 2. 3.

8. In a definite integral, you should find the antiderivative using substitution, change back to x, and then plug in endpoints. Ex. We could have changed the endpoints to u…

9. Integration by Parts To find our formula, we use the product rule (don’t write this down): -We pick u and dv -After using formula, you still have x’s

10. Ex.

11. Picking u and dv: You must be able to find the antiderivative of dv. Often, u will be a power of x.

12. Ex.

13. Pract. 1. 2. 3.