In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₁^∞ (1 / (x² + 3x)) dx

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Identify the type of improper integral: Since the upper limit is infinity, this is an improper integral of the form \(\int_1^{\infty} f(x) \, dx\) where the interval is unbounded.

Rewrite the integral with a limit to handle the improper nature: Express the integral as \(\lim_{t \to \infty} \int_1^t \frac{1}{x^2 + 3x} \, dx\).

Simplify the integrand using partial fraction decomposition: Factor the denominator as \(x(x + 3)\) and write \(\frac{1}{x(x+3)} = \frac{A}{x} + \frac{B}{x+3}\), then solve for constants \(A\) and \(B\).

Integrate the decomposed fractions: Integrate each term separately to get expressions involving natural logarithms, such as \(\int \frac{1}{x} \, dx = \ln|x|\) and \(\int \frac{1}{x+3} \, dx = \ln|x+3|\).

Evaluate the definite integral from 1 to \(t\), then take the limit as \(t\) approaches infinity to determine if the integral converges or diverges.

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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 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.

Partial Fraction Decomposition

Partial fraction decomposition breaks a rational function into simpler fractions that are easier to integrate. For example, expressing 1/(x² + 3x) as A/x + B/(x+3) allows straightforward integration of each term separately, facilitating the evaluation of the integral.

Limit Evaluation for Convergence

After integrating the function over a finite interval, we evaluate the limit as the upper bound approaches infinity. If this limit exists and is finite, the improper integral converges; otherwise, it diverges. This step is crucial to conclude the behavior of the integral over an infinite domain.

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