The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞² (2 dx) / (x² + 4)

Verified step by step guidance

1. Recognize that the integral is an improper integral because the lower limit is negative infinity. This means we need to express the integral as a limit: \(\displaystyle \int_{-\infty}^2 \frac{2}{x^2 + 4} \, dx = \lim_{a \to -\infty} \int_a^2 \frac{2}{x^2 + 4} \, dx\).
2. Recall the standard integral formula for \(\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C\). In this problem, \(a^2 = 4\), so \(a = 2\).
3. Rewrite the integrand to match the formula: \(\frac{2}{x^2 + 4} = 2 \cdot \frac{1}{x^2 + 2^2}\). This allows us to factor out the constant 2 and apply the formula.
4. Evaluate the definite integral from \(a\) to 2 using the antiderivative: \(2 \cdot \left[ \frac{1}{2} \arctan\left( \frac{x}{2} \right) \right]_a^2 = \left[ \arctan\left( \frac{x}{2} \right) \right]_a^2\).
5. Take the limit as \(a\) approaches negative infinity: \(\lim_{a \to -\infty} \left( \arctan\left( \frac{2}{2} \right) - \arctan\left( \frac{a}{2} \right) \right)\). Use the fact that \(\arctan(x)\) approaches \(-\frac{\pi}{2}\) as \(x \to -\infty\) to complete the evaluation.

Key Concepts

Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, we use limits to define the integral as the limit of a definite integral over a finite interval approaching the problematic point or infinity.

Integration of Rational Functions

Rational functions are ratios of polynomials. Integrals of the form ∫ dx / (x² + a²) are standard and result in inverse trigonometric functions, specifically arctangent, which helps in evaluating integrals without tables.

Limits and Convergence of Integrals

Determining whether an improper integral converges requires evaluating the limit of the integral as the bound approaches infinity. If the limit exists and is finite, the integral converges; otherwise, it diverges.