1. Trigonometric Substitutions
Section 8.4b

2. Trigonometric Substitutions
How do we evaluate this integral?
These trigonometric substitutions allow us to replace
binomials of the form
by single squared terms, and thereby transform a number
of integrals (like the one above) into ones we can evaluate
directly or find in a table of integrals.

3. Trigonometric Substitutions
The most common substitutions are based on the following
reference triangles:
With

4. Trigonometric Substitutions
The most common substitutions are based on the following
reference triangles:
With

5. Trigonometric Substitutions
The most common substitutions are based on the following
reference triangles:
With

6. Trigonometric Substitutions
replaces with
replaces with
replaces with
Also, we want any substitution to be reversible so we can
change back to the original variable afterward. For example:
requires
Essentially, keep positives
with any absolute values…
with

7. Trigonometric Substitutions
Set
Evaluate

8. Trigonometric Substitutions
Set
Evaluate

9. Trigonometric Substitutions
Evaluate
Appendix A7, Formula 88 (p.631)

10. Trigonometric Substitutions
Evaluate

11. Trigonometric Substitutions
Evaluate