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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 = − cos x + C cos x dx = sin x + C tan x dx = ln | sec x| + C cot x dx = ln | sin x| + C sec x dx = ln | sec x + tan x| + C csc x dx = − ln | csc x + cot x| + C Z Z Z Z Z Z π/3 2. cot 2xdx π/4 1 1 ln 3 − ln 2 4 2 Powers of Sines & Cosines: For each of the following, evaluate the given integral. Z 3. sin (x) cos3 (x) dx 1 − cos4 x + C 4 1 Z 4. sin3 (x) cos4 (x) dx 1 1 cos7 x − cos5 x + C 7 5 Z √ 5. sin x cos3 (x) dx 2 2 (sin x)3/2 − (sin x)7/2 + C 3 7 Z 6. sin2 x dx x 1 − sin (2x) + C 2 4 Z 7. sin3 (bx) dx, where b is a non-zero constant 1 1 cos3 (bx) − cos (bx) + C; Detailed Solution: Here 3b b Z 8. sin2 x cos2 x dx x 1 − sin (4x) + C 8 32 Z π/2 9. cos3 x dx π/4 √ 2 5 2 − 3 12 Z 10. cos4 5x dx 3 1 1 x+ sin (10x) + sin (20x) + C 8 20 160 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. sin (A − B) = sin A cos B − cos A sin B 2 (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 Adding the given identity to the identity from part (a) and then dividing both sides by 2 yields the desired result. 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. cos (A − B) = cos A cos B + sin A sin 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 Adding the given identity to the identity from part (a) and then dividing both sides by 2 yields the desired result. (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 Subtracting the given identity from the identity from part (a) and then dividing both sides by 2 yields the desired result. 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 1 5x 1 3x − cos − cos +C 5 2 3 2 Z 14. cos (3x) cos (4x) dx 1 1 sin x + sin (7x) + C 2 14 Z 15. sin (5x) cos (2x) dx 1 1 − cos (3x) − cos (7x) + C 6 14 3 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] 0 Powers of Tangents & Secants: For each of the following, evaluate the given integral. Z 17. tan2 3x dx −x + Z 18. 1 tan (3x) + C 3 π/4 tan3 (x) sec3 (x) dx 0 √ 2 1+ 2 15 Z 19. tan (x) sec3 (x)dx 1 sec3 x + C 3 Z 20. tan3 (x) sec4 (x) dx 1 1 tan6 x + tan4 x + C 6 4 Z 21. tan5 (2x) sec2 (2x) dx 1 tan6 (2x) + C 12 4 Z 22. tan (x) sec5/2 (x) dx 2 sec5/2 x + C; Detailed Solution: Here 5 Z 23. sec4 x dx 1 tan3 x + tan x + C 3 Z π 24. Consider sec x dx π/2 (a) Explain why this integral is improper. π The integral is improper because sec x has an infinite discontinuity at x = 2 which is the lower limit of integration. (b) Evaluate the given integral. If it diverges, explain why. Z π sec x dx = −∞ The integral diverges because π/2 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) 1 1 sec (x) tan (x) + ln | sec x + tan x| + C 2 2 Z (b) Use part (a) to evaluate tan2 (x) sec (x) dx 1 1 sec (x) tan (x) − ln | sec x + tan x| + C 2 2 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. √ 2−1 (b) Compute the volume of the solid which results from revolving R around the x-axis. π 2 5 π 3π 27. Find the length of the curve y = ln (sin x) on the interval , . 4 4 √ 2 ln 2 + 2 − ln 2; Detailed Solution: Here 6