Integration by Parts Lesson 8.2

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

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

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

Strategy Given an integral we 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

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 u du dv v x ex dx dx ex

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

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

Double Trouble Sometimes the second integral must also be done by parts u x2 du 2x dx dv sin x v -cos x u du dv v

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

Assignment Lesson 8.2A Page 531 Exercises 1 – 35 odd

dv and its anti-derivatives Tabular Method Consider Determine u, dv as before Set up table Quit when you get 0 in middle column Solution obtained by adding signed products of diagonal entries Alternating Signs u and its derivatives dv and its anti-derivatives + sin 4x - 2x 2

What is the disk thickness? 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?

Found this in your teacher's waste basket. Where Is the Error? Found this in your teacher's waste basket. He's indefinitely nuts!

Assignment Lesson 8.2B Page 532 Exercises 47 – 57 odd, 99, 101, 107