Day 14 in MAT229

Today:

• Announcements:
  • Your 6.6 section homework is due tomorrow

  • Sorry, I've not yet graded your labs.

• Today we feature Section 7.1: Integration by parts.
  • We'll motivate this topic by considering exercise #66, p. 494 in detail.

  • 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:
      

  • Examples:
    1. #9, p. 492
    2. #10
    3. #26
    4. #39
    5. #40
    6. #51