7. Antiderivatives & Indefinite Integrals

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (3x + 1) (4 - x) dx

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ y² (3y³ + 1)⁴ dy

7–64. Integration review Evaluate the following integrals.

49. ∫ √(9 + √(t + 1)) dt

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ 6 dt / (9t² + 1)²

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

46. ∫ (x³ + 4x² + 12x + 4)/((x² + 4x + 10)²) dx

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (x² - 36) / (x - 6) 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 − cot²x) 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.

∫x⁻¹ᐟ³ 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.

∫(8y − 2 / y¹ᐟ⁴) dy

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

68. ∫ dx / sqrt((x - 1)(3 - x))

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 + tan²θ)dθ (Hint:1 + tan²θ = sec²θ)

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

__

∫ ( 3√ t + 4/t² ) dt

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ 𝓍⁷ √(𝓍⁴ + 1d𝓍)

7–84. Evaluate the following integrals.

13. ∫ [1 / (eˣ √(1 – e²ˣ))] 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.

∫csc θ/(csc θ − sin θ) dθ