7. Antiderivatives & Indefinite Integrals

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

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

∫ sin(2θ) dθ / (1 + cos(2θ))²

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.                                                                                                                                                           

(c) ∫ sin 2𝓍 d𝓍 = 2 ∫ sin 𝓍 d𝓍 .

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.

∫ sin(θ) sin(2θ) sin(3θ) dθ

Evaluate the integrals in Exercises 41–60.

45. ∫tanh(x/7)dx

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

67. ∫tan⁴(3y) dy

69. Different substitutions

b. Evaluate ∫(tan x sec² x) dx using the substitution u=secx.

9–61. Trigonometric integrals Evaluate the following integrals.

47. ∫ (csc⁴x)/(cot²x) dx

Evaluate the integrals in Exercises 33–52.

∫ sec(x) tan²(x) dx

Use reduction formulas to evaluate the integrals in Exercises 41–50.

∫ 2 sin^2(t) sec^4(t) dt

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫(√1 + tan 2t) sec² 2t dt

Evaluating integrals Evaluate the following integrals.                                                                                                                                      

 ∫ 𝓍² cos 𝓍³ 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.

∫ (csc t sin 3t dt)

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.

∫ (dθ / cos θ - 1)

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ sin(2x) cos(4x) dx