1. In this section, we will look at integrating more complicated trigonometric expressions and introduce the idea of trigonometric substitution.

2. Symbolic antidifferentiation methods developed thus far (substitution, integration by parts, partial fractions) handle many classes of functions. Two classes not specifically covered yet: 1. powers of trig functions 2. roots of quadratic expressions

7. To do this algebraically, it is a messy integration by parts (potentially multiple times). Often, the reduction formula is just used:

8. Combining the above ideas, appropriate u substitutions, and/or using trigonometric identities, we can handle most integrals involving trigonometric functions raised to powers.

9. Find

11. We now turn our attention to integrands containing one of the following: We will often have to initially complete the square of a quadratic expression to get one of these forms.

12. We make the following substitutions: We work the “new” integral in t using various techniques studied, and then ultimately re- substitute back to x.

13. Find