7. Antiderivatives & Indefinite Integrals

9–61. Trigonometric integrals Evaluate the following integrals.

34. ∫ tan⁹x sec⁴x dx

9–61. Trigonometric integrals Evaluate the following integrals.

37. ∫ [sec⁴(lnθ)]/θ dθ

9–61. Trigonometric integrals Evaluate the following integrals.

38. ∫ tan⁵θ sec⁴θ dθ

9–61. Trigonometric integrals Evaluate the following integrals.

43. ∫ tan³(4x) dx

9–61. Trigonometric integrals Evaluate the following integrals.

45. ∫ sec²x tan¹ᐟ²x dx

9–61. Trigonometric integrals Evaluate the following integrals.

50. ∫ csc¹⁰x cot³x dx

9–61. Trigonometric integrals Evaluate the following integrals.

51. ∫ (csc²x + csc⁴x) dx

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