Integration by Parts Lesson 8.2 Review Product Rule • Recall definition of derivative of the product of two functions Dx f ( x) g ( x) f ( x) g '( x) g ( x) f '( x) • Now we will manipulate this to get f ( x) g '( x) Dx f ( x) g ( x) g ( x) f '( x) Manipulating the Product Rule • Now take the integral of both sides f ( x) g '( x) dx D f ( x) g ( x) dx g ( x) f '( x) dx x • Which term above can be simplified? • This gives us f ( x) g '( x) dx f ( x) g ( x) g ( x) f '( x) dx Integration by Parts f ( x) g '( x) dx f ( x) g ( x) g ( x) f '( x) dx • It is customary to write this using substitution u = f(x) v = g(x) du = f '(x) dx dv = g'(x) dx u dv u v v du Strategy x x e dx we split the • Given an integral integrand into two parts Note: a certain First part labeled u The other labeled dv amount of trial and error will happen in making this split • 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 u dv u v v du Making the Split • A table to keep things organized is helpful u dv x ex dx du dx v ex xe • Decide what will be the u and the dv • This determines the du and the v • Now rewrite u v v du x e e dx x x x dx 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 x ln x dx • Choose a u u du and dv dv v • Determine the v and the du • Substitute the values, finish integration 5 u v v du __________________ Double Trouble • Sometimes the second integral must also be done by parts x 2 sin x dx u x2 du 2x dx dv sin x v -cos x x cos x 2 x cos x dx 2 u du dv v Going in Circles • When we end up with the v du the same as we started with x e • Try sin x dx • Should end up with e x sin x dx e cos x e sin x e sin x dx x x x • Add the integral to both sides, divide by 2 2 e sin x dx e cos x e sin x x x x Application • Consider the region bounded by y = cos x, y = 0, x = 0, and x=½π • 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? Assignment • Lesson 8.2A • Page 531 • Exercises 1 – 35 odd • Lesson 8.2B • Page 532 • Exercises 47 – 57, 99 – 105 odd