7. Antiderivatives & Indefinite Integrals

65-68. Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals.

67. ∫tan⁴(3y) dy

7–84. Evaluate the following integrals.

41. ∫ cot^(3/2)x · csc⁴x dx

7–84. Evaluate the following integrals.

49. ∫ tan³x · sec⁹x dx

7–84. Evaluate the following integrals.

53. ∫ eˣ cot³(eˣ) dx

Evaluate the following integrals.

∫ (1 - cosx)/(1 + cosx) dx

67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:

sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]

sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]

cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]

68. ∫ sin(5x)sin(7x) dx

67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:

sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]

sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]

cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]

70. ∫ cos(x)cos(2x) dx

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

6. ∫ (2 − sin 2θ)/cos² 2θ dθ

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

22. ∫ tan³ 5θ dθ

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

29. ∫ cos⁴ x/sin⁶ x dx

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

32. ∫ csc²(6x) cot(6x) dx

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

48. ∫ sin(3x) cos⁶(3x) dx

7–84. Evaluate the following integrals.

82. ∫ 1/(1 + tanx) dx

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ dx / (1 - sin² x)

90–103. Indefinite integrals Determine the following indefinite integrals.

∫(1 + 3 cosΘ) dΘ