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.

∫(1/x² − x² − 1/3) 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.

∫(√x + ³√x) 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

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.

∫2x(1 − x⁻³) 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.

∫(t√t + √t) / t² dt

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

a. u = 1/(x + 1)

What is the value of the integral?

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

∫ (cos(√x))/(√x) dx

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

∫ 1/(x(ln(x))²) dx

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

∫ e√x / √x dx

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

∫ (sin⁻¹ x)² / √(1 - x²) dx

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

∫ (x⁴ - 1) / (x⁵ - 5x + 1) dx

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

∫ x / (x + √(x² + 2)) dx

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

∫ dx / (x √(x + 4))

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

∫ x (7x + 5)^(3/2) dx