Today we introduced Integration by Parts. Here’s one video example of I-by-P. And here is another longer video with a good explanation. We did these in-class examples:
\(\displaystyle \int x \sin x \, dx\).
\(\displaystyle \int \ln x \, dx\).
\(\displaystyle \int \arctan x \, dx\).
\(\displaystyle \int t^2 e^t \, dt\).
The last example requires integration by parts twice. An easier way to do integration by parts in this situation is tabular integration. We finished with one final tabular integration problem:
Here is a strange video of someone teaching the tabular method.
Today we talked about trigonometric integrals. Before starting, we did a classic tricky integration by parts problem. You can see a solution to this problem here.
Then we did several examples of trigonometric integrals. We followed the book pretty closely on this, so I’d recommend reading section 7.2. If you prefer videos, here’s a pretty good video explanation.
\(\displaystyle \int \cos x \sin^2 x \, dx\)
\(\displaystyle \int \cos^3 x \, dx\)
\(\displaystyle \int \tan^6 x \sec^4 x \, dx\)
\(\displaystyle \int \tan^3 x \sec^7 x \, dx\). Hint: let \(u = \sec x\). Keep a \(\sec x \tan x\) factor to become the \(du\).
\(\displaystyle \int \frac{ \sec x}{\tan^2 x} \,dx\). Hint switch everything to sines and cosines first.
All of these problems could be solved with u-substitution and the two basic trig identities:
Then we talked about the trigonometric product identities:
Then we used those identities to evaluate:
\(\displaystyle \int \cos^2 x \, dx\).
\(\displaystyle \int \sin(8x) \cos(6x) \, dx\).
Today we started with two example problems. First we used the ideas from Wednesday to compute
Then we showed that
Then we introduced the idea of trigonometric substitutions. Here is a video introduction. We did the following examples in class:
\(\displaystyle \int \frac{dx}{(x^2+1)^{3/2}}\).
\(\displaystyle \int \frac{\sqrt{x^2-9}}{x} \, dx\).
\(\displaystyle \int \frac{1}{25+x^2} \, dx\).