7. Antiderivatives & Indefinite Integrals

1. State the half-angle identities used to integrate sin²x and cos²x.

9–61. Trigonometric integrals Evaluate the following integrals.

10. ∫ sin³x dx

9–61. Trigonometric integrals Evaluate the following integrals.

11. ∫ sin²(3x) dx

9–61. Trigonometric integrals Evaluate the following integrals.

13. ∫ sin⁵x dx

7–84. Evaluate the following integrals.

27. ∫ sin⁴(x/2) dx

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

8. ∫ sin 3x cos 2x dx

9–61. Trigonometric integrals Evaluate the following integrals.

15. ∫ sin³x cos²x dx

9–61. Trigonometric integrals Evaluate the following integrals.

16. ∫ sin²θ cos⁵θ dθ

9–61. Trigonometric integrals Evaluate the following integrals.

20. ∫ sin⁻³ᐟ²x cos³x dx

9–61. Trigonometric integrals Evaluate the following integrals.

26. ∫ sin³x cos³ᐟ²x dx

9–61. Trigonometric integrals Evaluate the following integrals.

25. ∫ sin²x cos⁴x dx

9–61. Trigonometric integrals Evaluate the following integrals.

28. ∫ 6 sec⁴x dx

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

 (b) ∫ sec 5𝓍 tan 5𝓍 d𝓍

Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.                                                                                              

 ∫ 8𝓍 cos (4𝓍² + 3) d𝓍, u = 4𝓍² + 3

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

 (d) ∫ cos 𝓍/7 d𝓍