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Trigonometric Integral SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 7.3 of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes. EXPECTED SKILLS: • Know antiderivatives for all six elementary trigonometric functions. • Be able to evaluate integrals that involve powers of sine, cosine, tangent, and secant by using appropriate trigonometric identities. PRACTICE PROBLEMS: 1. Fill in the following table Z sin x dx = Z cos x dx = Z tan x dx = Z cot x dx = Z sec x dx = Z csc x dx = Z π/3 cot 2xdx 2. π/4 Powers of Sines & Cosines: For each of the following, evaluate the given integral. Z 3. sin (x) cos3 (x) dx Z 4. sin3 (x) cos4 (x) dx Z √ sin x cos3 (x) dx 5. 1 Z 6. Z 7. Z 8. Z sin2 x dx sin3 (bx) dx, where b is a non-zero constant sin2 x cos2 x dx π/2 cos3 x dx 9. π/4 Z 10. cos4 5x dx 11. Consider the trigonmetric identity sin (A + B) = sin A cos B + cos A sin B (a) Use this identity to derive an identity for sin (A − B) in terms of sin A, cos A, sin B, and cos B. (b) Use the given identity and your answer for part (a) to derive the following identity: sin A cos B = 1 [sin (A − B) + sin (A + B)] 2 12. Consider the trigonmetric identity cos (A + B) = cos A cos B − sin A sin B (a) Use this identity to derive an identity for cos (A − B) in terms of sin A, cos A, sin B, and cos B. (b) Use the given identity and your answer for part (a) to derive the following identity: cos A cos B = 1 [cos (A − B) + cos (A + B)] 2 (c) Use the given identity and your answer for part (a) to derive the following identity: sin A sin B = 1 [cos (A − B) − cos (A + B)] 2 For problems 13-16, use an appropriate identity from problem 11 or 12 to evaluate the given integral. Z x 13. sin (2x) cos dx 2 Z 14. cos (3x) cos (4x) dx Z 15. sin (5x) cos (2x) dx 2 16. The graph of f (x) = sin 2x sin 5x on the interval [−π, π] is shown below. Compute the net signed area between the graph of f (x) and the x-axis on the interval [−π, π] Powers of Tangents & Secants: For each of the following, evaluate the given integral. Z 17. tan2 3x dx Z 18. π/4 tan3 (x) sec3 (x) dx 0 Z 19. Z 20. Z 21. Z 22. Z 23. tan (x) sec3 (x)dx tan3 (x) sec4 (x) dx tan5 (2x) sec2 (2x) dx tan (x) sec5/2 (x) dx sec4 x dx Z π 24. Consider sec x dx π/2 (a) Explain why this integral is improper. (b) Evaluate the given integral. If it diverges, explain why. 3 Z 25. (a) Use integration by parts to evaluate sec3 (x) dx. (Hint: sec3 x = sec2 x sec x and tan2 x = sec2 x − 1) Z (b) Use part (a) to evaluate tan2 (x) sec (x) dx 26. Let R beh the iregion bounded between the graphs of y = sin x and y = cos x on the π π , . interval 4 2 (a) Compute the area of R. (b) Compute the volume of the solid which results from revolving R around the x-axis. π 3π 27. Find the length of the curve y = ln (sin x) on the interval , . 4 4 4