Day 13 in Mat229

• M-R, 10-3
• F, 10-noon
• And it's right by my office, so you can just drop in!

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

• On problem 2, 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 requires 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 $\tan^{-1}(x)$).

• 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.

Let's look at a few examples which illustrate the importance of trigonometric identities.

• Checkpoint 3.6: \[ \int \cos^{3}(x)\sin^{2}(x) d x \]
• Checkpoint 3.7: \[ \int \cos^{2}(x) d x \]

• 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: \[ \cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\beta)\sin(\alpha) \]