7. Antiderivatives & Indefinite Integrals

4. Is a reduction formula an analytical method or a numerical method? Explain.

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.

∫(−3csc²x)dx

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

39. ∫ x²/(100 - x²)^(3/2) dx

Use any method to evaluate the integrals in Exercises 55–66.

∫ (x⁴ - 1) / (x⁵ - 5x + 1) dx

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

54. ∫ y⁴/(1 + y²) dy

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

∫ sec Θ(tan Θ + sec Θ + cos Θ)dΘ

Find the following indefinite integral.

$\(\int\)4x^3dx$

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

44. ∫ 1/√(16 + 4x²) 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 + ³√x) dx

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

∫ (1/2y)dy

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

∫ x dx / √(2 − x)

Is either of the following equations correct? Give reasons for your answers.

a. (1/cosx) ∫ cos x dx = tan x + C

b. (1/cosx) ∫ cos x dx = tan x + C / cos x

Evaluating Indefinite Integrals

Evaluate the integrals in Exercises 37–46.

∫ 2(cos x)⁻¹/² sin x dx

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (12/x)dx

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

∫ √(x² - 4) / x dx