Monday 25 September class UPDATED

(To be updated – I am typing up notes and quiz solutions and it will take a bit of time.)

Topics:

• More Trigonometric Integrals:

Here are partial notes on the examples I discussed in class (most of them). They prompt you to complete certain parts of the integration, and it will help you prepare for Test 1 if you do this yourself! (Without trying to look up a solution somewhere, that is!)

Here are the relevant videos from PatrickJMT which I have previously linked:

Trignonometric integrals part 3 of 6 

(integrals of the form \int \tan^{m}(x)\sec^{n}(x)\textrm{d}x)

Trignonometric integrals part 4 of 6 

(More examples of \int \tan^{m}(x)\sec^{n}(x)\textrm{d}x)

 

The main thing to remember in these types of integrals is that you want (if possible) to end up using substitution with u = \tan(x), so you want to end up with \sec^{2}(x) in your integrand in order to have \textrm{d}u = \sec^{2}(x)\textrm{d}x.

 

 

Trignonometric integrals part 5 of 6 

(Integrating \int \sin(mx)\cos(nx)\textrm{d}x)

 

In these we make use of the product-to-sum identities, which may be new to most of us!

\sin(mx)\sin(nx) = \frac{1}{2}\left[\cos(mx-nx) - \cos(mx+nx)\right] \sin(mx)\cos(nx) = \frac{1}{2}\left[\sin(mx-nx) - \sin(mx+nx)\right] \cos(mx)\cos(nx) = \frac{1}{2}\left[\cos(mx-nx) + \cos(mx+nx)\right]

 

And you may want to view this one also

Trignonometric integrals part 6 of 6 

(More examples of different kinds of trig integrals that don’t fit neatly into the above categories)

 

• Trigonometric substitution (introduction)

Note to follow…

 

Homework:

• Study and complete the examples when I post them (or from your notes).

• Do the WeBWorK: In the “Trigonometric Integrals” homework you should at least complete the ones that are powers of sines times powers of cosines by Tuesday night, since that is on Test 1. Also do the new “Trigonometric Substitution 1” assignment, only 2 problems, due by Tuesday night. Do not wait to the last minute!

• Don’t forget that Test 1 is scheduled for Wednesday. See the separate post for more information and review.

 

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!