Integration by parts
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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.

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Presentation Transcript

Review product rule
Review Product Rule

  • Recall definition of derivative of the product of two functions

  • Now we will manipulate this to get


Manipulating the product rule
Manipulating the Product Rule

  • Now take the integral of both sides

  • Which term above can be simplified?

  • This gives us


Integration by parts1
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
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

x

dx

ex dx

ex

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


Strategy hint
Strategy Hint

  • Trick is to select the correct function for u

  • A rule of thumb is the LIATE hierarchy ruleThe u should be first available from

    • Logarithmic

    • Inverse trigonometric

    • Algebraic

    • Trigonometric

    • Exponential


Try this
Try This

  • Given

  • Choose a uand dv

  • Determinethe v and the du

  • Substitute the values, finish integration


Double trouble
Double Trouble

  • Sometimes the second integral must also be done by parts


Going in circles
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


Application
Application

  • Consider the region bounded by y = cos x, y = 0, x = 0, and x = ½ π

  • What is the volume generated by rotatingthe region around the y-axis?

What is the radius?

What is the disk thickness?

What are the limits?


Assignment
Assignment

  • Lesson 8.2A

  • Page 531

  • Exercises 1 – 35 odd

  • Lesson 8.2B

  • Page 532

  • Exercises 47 – 57, 99 – 105 odd


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