In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₁^∞ (1 / x^(1/5)) dx

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Identify the type of improper integral: Since the upper limit of integration is infinity, this is an improper integral of the form \(\int_1^{\infty} \frac{1}{x^{1/5}} \, dx\).

Rewrite the integrand using exponent notation: \(\frac{1}{x^{1/5}}\) can be written as \(x^{-1/5}\).

Set up the integral with a limit to handle the improper integral: \(\lim_{t \to \infty} \int_1^t x^{-1/5} \, dx\).

Find the antiderivative of \(x^{-1/5}\): Use the power rule for integration, which states \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) for \(n \neq -1\). Here, \(n = -\frac{1}{5}\), so the antiderivative is \(\frac{x^{4/5}}{4/5} = \frac{5}{4} x^{4/5}\).

Evaluate the definite integral from 1 to \(t\) and then take the limit as \(t\) approaches infinity: Calculate \(\lim_{t \to \infty} \left[ \frac{5}{4} t^{4/5} - \frac{5}{4} \cdot 1^{4/5} \right]\) to determine if the integral converges or diverges.

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Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, we take limits of definite integrals as the interval approaches infinity or the point of discontinuity. Determining convergence means checking if this limit exists and is finite.

p-integral Test

The p-integral test helps determine convergence of integrals of the form ∫₁^∞ 1/x^p dx. If p > 1, the integral converges; if p ≤ 1, it diverges. This test is essential for quickly assessing the behavior of power function integrals over infinite intervals.

Evaluating Limits of Integrals

To evaluate an improper integral, express it as a limit of a definite integral with an upper bound t approaching infinity. Compute the antiderivative, substitute the limits, and then take the limit as t → ∞. This process confirms convergence and provides the integral's value if it converges.

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.∫₁^∞ (1 / x^(1/5)) dx