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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.8.6
Chapter 8, Problem 8.8.6

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋₈¹ dx / x^(1/3)

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1
Identify the integral to be evaluated: \(\int_{-8}^{1} \frac{dx}{x^{1/3}}\).
Rewrite the integrand using exponent rules: \(\frac{1}{x^{1/3}} = x^{-1/3}\), so the integral becomes \(\int_{-8}^{1} x^{-1/3} \, dx\).
Use the power rule for integration, which states that for \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\). Here, \(n = -\frac{1}{3}\), so \(n + 1 = \frac{2}{3}\).
Apply the power rule to get the antiderivative: \(\int x^{-1/3} \, dx = \frac{x^{2/3}}{2/3} + C = \frac{3}{2} x^{2/3} + C\).
Evaluate the definite integral by substituting the limits: calculate \(\left. \frac{3}{2} x^{2/3} \right|_{-8}^{1}\), which means compute \(\frac{3}{2} (1)^{2/3} - \frac{3}{2} (-8)^{2/3}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Improper Integrals

Improper integrals involve integrands with infinite discontinuities or infinite limits. When the integrand is undefined or unbounded at a point within the interval, the integral is evaluated as a limit approaching that point to determine convergence.
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Improper Integrals: Infinite Intervals

Integration of Power Functions

Integrating power functions of the form x^n involves using the formula ∫x^n dx = (x^(n+1))/(n+1) + C, valid for all n ≠ -1. This rule helps evaluate integrals where the integrand is a power of x, including fractional exponents.
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Representing Functions as Power Series

Convergence of Integrals with Negative and Fractional Exponents

When integrating functions like x^(m/n) over intervals including zero or negative values, it is crucial to check if the integral converges. For fractional exponents, the function may be undefined or discontinuous at zero, requiring careful limit evaluation.
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Integration Using Partial Fractions
Related Practice
Textbook Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

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Textbook Question

In Exercises 67–73, use integration by parts to establish the reduction formula.

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Textbook Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

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Textbook Question

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(Hint: Multiply by √((1 - sin(x)) / (1 - sin(x))))

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Textbook Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

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