1. Drill: Find dy/dx y = x 3 sin 2x y = e 2x ln (3x + 1) y = tan -1 2x Product rule: x 3 (2cos 2x) + 3x 2 sin (2x) 2x 3 cos 2x + 3x 2 sin (2x) Product Rule e 2x (3/(3x +1)) + 2e 2x ln (3x + 1) 3e 2x /(3x +1) + 2e 2x ln (3x + 1)

2. Antidifferentiation by Parts Lesson 6.3

3. Objectives Students will be able to: – use integration by parts to evaluate indefinite and definite integrals. – use rapid repeated integration or tabular method to evaluate indefinite integrals.

4. Integration by Parts Formula A way to integrate a product is to write it in the form If u and v are differentiable function of x, then

5. Example 1 Using Integration by Parts Evaluate

6. Example 1 Using Integration by Parts Evaluate

7. Example 1 Using Integration by Parts Evaluate

8. Example 2 Repeated Use of Integration by Parts Evaluate

10. Example 2 Repeated Use of Integration by Parts Evaluate

12. Example 3 Solving an Initial Value Problem Solve the differential equation dy/dx = xlnx subject to the initial condition y = -1 when x = 1 It is typically better to let u = lnx

14. Drill Solve the differential equation: dy/dx = x 2 e 4x (This means you will need to find the anti-derivative of dy/dx = x 2 e 4x )

16. Example 4 Solving for the unknown integral

18. Rapid Repeated Integration by Parts AKA: The Tabular Method Choose parts for u and dv. Differentiate the u’s until you have 0. Integrate the dv’s the same number of times. Multiply down diagonals. Alternate signs along the diagonals.

19. Example 5Rapid Repeated Integration by Parts Evaluate u and its derivativesdv and its integrals

20. Example 5Rapid Repeated Integration by Parts Evaluate

21. Example 5Rapid Repeated Integration by Parts Evaluate u and its derivativesdv and its integrals

22. Example 5Antidifferentiating ln x

23. Example 6Antidifferentiating sin -1 x

25. Homework Page 346/7: Day #1: 1-15 odd Page 347: 17-24