Integration by Parts

1 / 12

# Integration by Parts - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Integration by Parts' - fathia

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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

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
• 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
• Given
• Choose a uand dv
• Determinethe v and the du
• Substitute the values, finish integration
Double Trouble
• Sometimes the second integral must also be done by parts
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
• 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?