Day 14 in MAT229

Today:

• Announcements:
  • I hope that the exam went well for you. I've graded the first four problems, and, for the most part, people seem to be doing pretty well. I hope to have it finished by tomorrow.

  • Today we will look over section 7.2: Trigonometric Integrals
    • But we'll start with a little more practice on integration by parts. Let's take another look at the worksheet from Friday:
      • Generally I did notice lots of dx dropping, and some confusion between anti-differentiation and differentiation.

      • Problem 1 was a fairly straightforward integration by parts. Just a little clunky, due to the constants that appeared.

      • On problem 2, most of you missed the point that you could use your result from problem 1 to "instantly" write down the answer to the problem (using $n=0$): \[ \int \ln(x)\ dx = \int x^0\ln(x)\ dx = \frac{x^{0+1}\ln(x)}{(0+1)}-\frac{x^{0+1}}{(0+1)^2}+C = x \ln(x) - x + C \]

      • Problem 3 required two successive integrations-by-parts

      • Problem 4 didn't necessarily require the substitution, but that's a nice trick sometimes ($u=x^2$).

        Let's take a look at that one.

      • Problem 5 involved identifying a product including 1 (as did our example of $tan^{-1}$).

        And, by the way, as I wrote that "arctan" with what appears to be a negative exponent, I want to apologize on behalf of all mathematicians. Our notation is confusing. But $tan^{-1}$ (arctan) has nothing to do with the multiplicative reciprocal of $\tan$ (which is $\cot$, by the way!). Clearly arctan and cot are different....

      • Can you generalize the rule for Problem 6? Let's look at that one.

      • Problem 7 involves using integration by parts twice, to get back to where you started (but in a useful way). Let's check that out.
      I need to change the format a little, so that I get coherent output from each group. But, as I mentioned last time, I'm learning about some issues from walking around and answering questions, and that's good.

  • We'll look at a few examples which illustrate the importance of trigonometric identities.

  • We'll also 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 essential (and you can get two of them using texpand on your TI calculator). You can derive the rest from these three:
    • The trig form of the Pythagorean Theorem: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
    • Sine of a sum: \[ \sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+\sin(\beta)\cos(\alpha) \]
    • Cosine of a sum: \[ \cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\beta)\sin(\alpha) \]

  • So let's have a look at our handout for 7.2.