In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁴ [x / (x² + 9)^(2/5)] dx

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Identify the type of improper integral: Since the integral has a lower limit of negative infinity, it is an improper integral due to an infinite limit of integration.

Rewrite the integral as a limit to handle the improper nature: Express the integral as \(\lim_{t \to -\infty} \int_{t}^{4} \frac{x}{(x^{2} + 9)^{2/5}} \, dx\).

Consider the behavior of the integrand as \(x \to -\infty\): Analyze the function \(\frac{x}{(x^{2} + 9)^{2/5}}\) to determine if the integral converges by comparing it to a simpler function whose integral behavior is known.

Find the antiderivative of the integrand: Use substitution methods, such as letting \(u = x^{2} + 9\), to find an expression for the indefinite integral \(\int \frac{x}{(x^{2} + 9)^{2/5}} \, dx\).

Evaluate the definite integral using the antiderivative and then take the limit as \(t \to -\infty\): Substitute the limits into the antiderivative expression and analyze the limit 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 use limits to approach the problematic points, determining if the integral converges (has a finite value) or diverges (does not).

Convergence Tests for Improper Integrals

To decide if an improper integral converges, we analyze the behavior of the integrand near infinity or discontinuities. Techniques include comparison tests and evaluating limits of the integral's partial sums to check if they approach a finite number.

Integration Techniques for Rational Functions with Exponents

Integrals involving rational functions with fractional exponents often require substitution or algebraic manipulation. Recognizing patterns and applying appropriate substitutions simplifies the integral, making it easier to evaluate or determine convergence.

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.∫₋∞⁴ [x / (x² + 9)^(2/5)] dx