Evaluate the improper integrals in Exercises 53–62.
∫ from −∞ to ∞ of (1 / (4x² + 9)) dx

Verified step by step guidance

1. Recognize that the integral is an improper integral because the limits of integration are from \(-\infty\) to \(\infty\). This means we need to evaluate the integral as a limit: \(\int_{-\infty}^{\infty} \frac{1}{4x^{2} + 9} \, dx = \lim_{A \to \infty} \int_{-A}^{A} \frac{1}{4x^{2} + 9} \, dx\).
2. Identify the integrand as a rational function that resembles the form \(\frac{1}{x^{2} + a^{2}}\), which suggests using a standard integral formula: \(\int \frac{dx}{x^{2} + a^{2}} = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C\).
3. Rewrite the denominator to match the standard form by factoring out the constant: \(4x^{2} + 9 = (2x)^{2} + 3^{2}\). Let \(u = 2x\), so \(du = 2 dx\) or \(dx = \frac{du}{2}\).
4. Substitute into the integral: \(\int \frac{1}{u^{2} + 3^{2}} \cdot \frac{du}{2} = \frac{1}{2} \int \frac{du}{u^{2} + 3^{2}}\). Then apply the arctangent formula to get \(\frac{1}{2} \cdot \frac{1}{3} \arctan\left(\frac{u}{3}\right) + C = \frac{1}{6} \arctan\left(\frac{2x}{3}\right) + C\).
5. Evaluate the definite integral from \(-A\) to \(A\) and then take the limit as \(A \to \infty\): \(\lim_{A \to \infty} \left[ \frac{1}{6} \arctan\left(\frac{2A}{3}\right) - \frac{1}{6} \arctan\left(\frac{-2A}{3}\right) \right]\). Use the fact that \(\arctan(x)\) approaches \(\frac{\pi}{2}\) as \(x \to \infty\) and \(-\frac{\pi}{2}\) as \(x \to -\infty\) to simplify.

Key Concepts

Improper Integrals

Improper integrals involve integration over an infinite interval or where the integrand has an infinite discontinuity. To evaluate them, the integral is expressed as a limit, approaching the infinite bound or the point of discontinuity, ensuring convergence or divergence is determined.

Integration of Rational Functions

Rational functions are ratios of polynomials. Integrating them often involves algebraic manipulation or recognizing standard integral forms, such as those leading to inverse trigonometric functions. For example, integrals of the form 1/(x² + a²) relate to arctangent functions.

Use of Limits in Definite Integrals

When evaluating definite integrals with infinite limits, limits are used to define the integral properly. The integral from −∞ to ∞ is split into two limits, each evaluated separately, and their sum is considered to determine if the integral converges to a finite value.