1. 8.4
Trigonometric Substitutions

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.
It’s purpose is to re-write integrands as trig functions that we can work with.
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. 3
We can get into the necessary form by completing the square.

8. 4
Complete the square:

9. Here are a couple of shortcuts that are the result of Trigonometric Substitution:
Haven’t we done these
before?

10. 5

11. 6

12. 7
We also need to consider the limits of integration: