1. Section 8.3 – Trigonometric Integrals

2. Rewriting tools for Integrating: Trigonometric Identities
Use these identities to help evaluate integrals with trigonometric functions.

3. Rewriting tools for Integrating: Pythagorean Identities
Use these identities to help evaluate integrals with trigonometric functions.

4. Rewriting tools for Integrating: Double Angle Formulas
Use these identities to help evaluate integrals with trigonometric functions.

5. Rewriting tools for Integrating: Power-Reducing Formulas
Use these identities to help evaluate integrals with trigonometric functions.

6. Goal of Trigonometric Integrals
If needed, rewrite the integral into a form that can be integrated.

7. Example 1
Evaluate: sin 3 𝑥 𝑑𝑥

8. Example 2
Evaluate: sin 4 𝑥 cos 5 𝑥 𝑑𝑥

9. Example 3
Evaluate: sin 4 𝑥 𝑑𝑥

10. Example 4
Evaluate: sin 2 𝑥 cos 4 𝑥 𝑑𝑥

11. White Board Challenge
Combine Integration by Parts and Trigonometric Identities to evaluate: sec 3 𝑥 𝑑𝑥
Pick the u and dv.
Find du and v.
Apply the formula.

12. White Board Challenge OR

13. White Board Challenge Evaluate: sec 3 𝑥 𝑑𝑥 Pick the u and dv.
Find du and v.

14. Example 5 Evaluate: sec 5 𝑥 𝑑𝑥 Pick the u and dv. Find du and v.
Apply the formula.

15. Example 6
Evaluate: tan 2 𝑥 sec 3 𝑥 𝑑𝑥

16. Example 7
Evaluate: 1− tan 2 𝑥 sec 2 𝑥 𝑑𝑥

17. Example 7 OR

18. Example 7
Evaluate: 1− tan 2 𝑥 sec 2 𝑥 𝑑𝑥

19. Example 8
Evaluate: 𝑑𝑥 cos 𝑥 −1

20. Example 9
Evaluate: cos 2 sin 𝑥 cos 𝑥 𝑑𝑥
Substitution:

21. General Guidelines for Trig Integrals
For sin 𝑛 𝑥 cos 𝑚 𝑥 𝑑𝑥 we have the following:
n odd.
Strip one sine out and convert the rest of the integral to only cosines using sin 2 𝑥 =1− cos 2 𝑥 . Then use the substitution with 𝑢= cos 𝑥 .
m odd.
Strip one cosine out and convert the rest of the integral to only sines using cos 2 𝑥 =1− sin 2 𝑥 . Then use the substitution with 𝑢= sin 𝑥 .
n and m both odd.
Use either 1. or 2.
n and m both even.
Use double angle, half angle, and/or power reducing formulas to reduce the integral into a form that can be integrated.
Try to still follow the rules even if m and n are negative AND count zero as even.

22. General Guidelines for Trig Integrals
For tan 𝑛 𝑥 sec 𝑚 𝑥 𝑑𝑥 we have the following:
n odd.
Strip one tangent and one secant out and convert the rest of the integral to only secants using tan 2 𝑥 = sec 2 𝑥 −1. Then use the substitution with 𝑢= sec 𝑥 .
m even.
Strip two secants out and convert the rest of the integral to only tangents using sec 2 𝑥 =1+ tan 2 𝑥 . Then use the substitution with 𝑢= tan 𝑥 .
n odd and m even.
Use either 1. or 2.
n even and m odd.
Each integral will be dealt with differently. Try Trigonometric Identities, Substitution, or Integration by Parts.
Try to still follow the rules even if m and n are negative AND count zero as even.