- Your 6.6 section homework is due today.
- Your quiz opportunity today is over section 7.2.
- The log lab is graded.
- Yesterday I used a problem from physics to
motivate
this topic (exercise #66, p. 494).
We noticed the value of our calculators, of course, but were able to use a bit of a "trick" to solve the problem (to calculate the integral
)
- Derivation of the integration by parts formula:
- First of all, the big picture: integration by parts is just the
product rule backwards. This integration technique, like all
integration techniques, is really just a differentiation technique in
reverse.
- Integration by parts may need to be carried out multiple times:
sometimes the idea is to simplify the integral each time, until a
really simple one arises allowing us to calculate the final solution
(e.g. Example 6, p. 491). Sometimes it's something of a trick: we
compute the integral multiple times in order to return to the original
integral, allowing us to solve an equation for the original integral
(e.g. Example 4, p. 490).
- Suggested strategy:
- Choose u so that u' is simpler than u itself.
- Choose v' so that
can be evaluated.
- Sometimes v'=1 is a good choice.
- Of course, all this works with definite integrals:
- First of all, the big picture: integration by parts is just the
product rule backwards. This integration technique, like all
integration techniques, is really just a differentiation technique in
reverse.
- Examples:
- #1, p. 492
- #3
- #9
- #10
- #26
- #39
- #40
- #51
- We'll look at a few examples which illustrate the importance of
trigonometric identities
- #65, 500
- #1
- #6
- Now let's look at some of the useful identities that we'll want to
use from time to time. The good news is that only three are really
necessary (and you can get two of them using texpand on your
calculator):
- The trig form of the Pythagorean Theorem
- Sine of a sum
- Cosine of a sum