7. Antiderivatives & Indefinite Integrals

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ cos(θ / 2) cos(7θ) dθ

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ 𝓍 sin 𝓍² cos⁸ 𝓍² d𝓍

9–61. Trigonometric integrals Evaluate the following integrals.

26. ∫ sin³x cos³ᐟ²x dx

9–61. Trigonometric integrals Evaluate the following integrals.

15. ∫ sin³x cos²x dx

Evaluate the integrals in Exercises 39–56.

49. ∫3sec²t/(6 + 3tan(t)) dt

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

Use any method to evaluate the integrals in Exercises 65–70.

∫ cot(x) / cos²(x) dx

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (tan θ + 3 / sin θ) dθ

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

 ∫ sec² (10𝓍 + 7) d𝓍

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             

 ∫ sin² 𝓍 d𝓍

Evaluate the integrals in Exercises 33–52.

∫ cot³(t) csc⁴(t) dt

4. Describe the method used to integrate sinᵐx cosⁿx, for m even and n odd.

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ (2 − cosx + sinx) / sin²x dx

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

 ∫ 𝓍 csc 𝓍² cot 𝓍² d𝓍

68. Different methods

a. Evaluate ∫(cot x csc² x) dx using the substitution u=cotx.