Overview
We shall very soon see that a special type of substitution leads to integrals that only contain trigonometric functions. Therefore, we start by looking at different types of integrals that involve trigonometric functions. We will focus our attention on those types of integrals where a stepbystep procedure can be applied to find the antiderivative. There are quite a few videos to watch to prepare for this section, each one covering a specific type of integral.
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 and memorize the common trigonometric identity \(\sin^2 x + \cos^2 x = 1\).

Understand which substitution to use for integrals of the type \(\int \sin^n x \cos^m x\;dx\) when either \(n\) or \(m\) is odd.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Understand how the identities \(\tan^2 x + 1 = \sec^2 x\) and \(1+\cot^2 x = \csc^2 x\) can be obtained from the identity \(\sin^2 x + \cos^2 x = 1\).

Use appropriate substitutions and trigonometric identities to evaluate integrals of the type
 \(\int \sin^n x \cos^m x\;dx\) (all cases)
 \(\int \tan^n x \sec^m x\;dx\) (m is even or n is odd)
 \(\int \sin(nx) \cos(mx)\;dx\)
 \(\int \cos(nx) \sin(mx)\;dx\)
 \(\int \cos(nx) \cos(mx)\;dx\)

\(\int \sin(nx) \sin(mx)\;dx\)

Be prepared to evaluate trigonometric integrals that do not fit any of the previous cases.
To prepare for class

Watch the following video which explains how to deal with integrals of the type \(\int \sin^n x \cos^m x\;dx\), where either \(m\) or \(n\) is odd:

Do Problem 1 of the Preview Activity for this section (on WeBWorK if required by your teacher).

Watch the following video which explains how to deal with integrals of the type \(\int \sin^n x \cos^m x\;dx\), where both \(m\) and \(n\) are even:

Do Problem 2 of the Preview Activity for this section (on WeBWorK if required by your teacher).

Watch the following video which explains how to deal with integrals of the type \(\int \tan^n x \sec^m x\;dx\), where \(m\) is even:

Watch the following video which explains how to deal with integrals of the type \(\int \tan^n x \sec^m x\;dx\), where \(n\) is odd:

Do Problem 3 of the Preview Activity for this section (on WeBWorK if required by your teacher).

Watch the following video which explains how to deal with integrals of the type \(\int \sin(nx) \cos(mx)\;dx\):

Do Problem 4 of the Preview Activity for this section (on WeBWorK if required by your teacher).

Watch the first example (you may watch the other examples if you like) of this video which suggests that if the trigonometric integral does not fit any of the previous cases, we may want to try first expressing it in terms of \(\sin x\) and \(\cos x\) only.