7. Antiderivatives & Indefinite Integrals

Evaluate the integrals in Exercises 33–52.

∫ sec³(x) tan³(x) dx

9–61. Trigonometric integrals Evaluate the following integrals.

43. ∫ tan³(4x) dx

Evaluate the integrals in Exercises 33–52.

∫ sec⁶(x) dx

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

∫ sin(t / 3) sin(t / 6) dt

Evaluate the integrals in Exercises 33–52.

∫ sec⁴(x) tan²(x) dx

9–61. Trigonometric integrals Evaluate the following integrals.

31. ∫ 20 tan⁶x dx

Evaluate the integrals in Exercises 41–60.

47. ∫sech²(x - 1/2)dx

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

∫ 8 cos^4(2πt) dt

Evaluate the integrals in Exercises 1–22.

∫ cos³(2x) sin⁵(2x) dx

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

∫ cos t dt / (1 - cos t)

Evaluate the integrals in Exercises 41–60.

49. ∫(sech(√t)tanh(√t)dt)/√t

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

∫ sin³(x) / cos⁴(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.

∫ (sec t + cot t)² dt

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

∫ sin³(θ) cos(2θ) dθ

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