1. Integration by Parts Lesson 8.2

2. Review Product Rule Recall definition of derivative of the product of two functions Now we will manipulate this to get

3. Manipulating the Product Rule Now take the integral of both sides Which term above can be simplified? This gives us

4. Integration by Parts It is customary to write this using substitution  u = f(x)du = f '(x) dx  v = g(x) dv = g'(x) dx

5. Strategy Given an integralwe split the integrand into two parts  First part labeled u  The other labeled dv Guidelines for making the split  The dv always includes the dx  The dv must be integratable  v du is easier to integrate than u dv Note: a certain amount of trial and error will happen in making this split

6. Making the Split A table to keep things organized is helpful Decide what will be the u and the dv This determines the du and the v Now rewrite udu dvv x e x dx dx exex

7. Strategy Hint Trick is to select the correct function for u A rule of thumb is the LIATE hierarchy rule The u should be first available from  Logarithmic  Inverse trigonometric  Algebraic  Trigonometric  Exponential

8. Try This Given Choose a u and dv Determine the v and the du Substitute the values, finish integration udu dvv

9. Double Trouble Sometimes the second integral must also be done by parts udu dvv ux2x2 du2x dx dvsin xv-cos x

10. Going in Circles When we end up with the the same as we started with Try Should end up with Add the integral to both sides, divide by 2

11. Application Consider the region bounded by y = cos x, y = 0, x = 0, and x = ½ π What is the volume generated by rotating the region around the y-axis? What is the radius? What is the disk thickness? What are the limits? What is the radius? What is the disk thickness? What are the limits?

12. Assignment Lesson 8.2A Page 531 Exercises 1 – 35 odd Lesson 8.2B Page 532 Exercises 47 – 57, 99 – 105 odd