We now study the last advanced method for integration for this course, the method of Trigonometric Substitution: This unusual substitution introduces trigonometric functions into integrals which actually don't seem to involve trigonometric functions, but which cannot be (easily) evaluated using other methods. After applying this substitution, we (hopefully) end up with a trigonometric integral (i.e. a product of trigonometric functions) which we have discussed previously.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.
Review the basic trigonometric identities \(\sin^2 x+\cos^2 x = 1\), \(\tan^2 x + 1 = \sec^2 x\), and \(1+\cot^2 x = \csc^2 x\).
Recognize terms in a given integrand which lend themselves to trigonometric substitution, and decide which one to use:
- for a term of the form \(a^2-x^2\), use \(x=a \sin \theta\)
- for a term of the form \(a^2+x^2\), use \(x=a \tan \theta\)
- for a term of the form \(x^2-a^2\), use \(x=a \sec \theta\)
Apply an appropriate trigonometric substitution to a variety of integrals.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
Apply an appropriate trigonometric substitution to more difficult integrals.
Understand how the so-called "Weierstrass Substitution" or "Tangent half-angle" substitution (\(t=\tan(x/2)\)) can be seen as a form of "reverse trigonometric substitution" to remove trigonometric functions, and how to use it to integrate rational functions of sine and cosine.
To prepare for class
Watch the following videos which explain how to use Trigonometric Substitution:
Watch the following videos which explain how to use the so-called "Weierstrass substitution" \(t=\tan(x/2)\) to integrate rational functions of sine and cosine (note that this is the substitution in question 4 of the Lab Exercises sheet for 5.5b):
(optional) Watch the following videos to review the most important trig identities (and how to remember them) and some facts about the unit circle (including special values):