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

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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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주요 개념

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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.

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.

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.

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