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

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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

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

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?

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

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