1. Trigonometric Substitutions
Ch.8 Extra Topic:
Trigonometric Substitutions
Monticello (Thomas Jefferson’s home), Charlottesville, VA
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2004

2. We can use right triangles and the pythagorean theorem to simplify some problems.1
These are in the same form.

3. We can use right triangles and the pythagorean theorem to simplify some problems.1
This is a constant.

4. This method is called Trigonometric Substitution.
If the integral contains , we use the triangle at right.
If we need , we
move a to the hypotenuse.
If we need , we
move x to the hypotenuse.

5. 2
double angle formula

6. 2
double angle formula

7. 5
We can get into the necessary form by completing the square.

8. 6
Complete the square:

9. Here are a couple of shortcuts that result from Trigonometric Substitution:
These are on your list of formulas. They are not really new. p