7. Antiderivatives & Indefinite Integrals

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(−2cost) dt

60–69. Completing the square Evaluate the following integrals.

62. ∫ du / (2u² - 12u + 36)

57. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. The integral ∫ dx/(x² + 4x + 9) cannot be evaluated using a trigonometric substitution.

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

119. ∫ x³ / (1 + x²) dx

Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.

∫ [x / √(4 + x²)] dx

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(1/x² − x² − 1/3) dx

Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.

∫ [t / √(4t² − 1)] dt

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.

15. ∫ x / √(4x + 1) dx

Evaluate the integrals in Exercises 77–90.

77. ∫dx/√(-x²+4x-3)

Variations on the substitution method Evaluate the following integrals.                                                                                                        

 ∫ (𝒵 + 1) √(3𝒵 + 2) d𝒵

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.

31. ∫ √(x² - 8x) dx, x > 8

Evaluate the indefinite integral as an infinite series: $\int arctan\left(x^4\right) dx$

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

74. ∫ dx/√(√(1 + √x))

For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.

∫ √(x² + 2x + 2) / (x² + 2x + 1) dx

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(4secx tanx − 2 sec²x)dx