The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ dx / [(x + 1)(x² + 1)]

Verified step by step guidance

1. Start by expressing the integrand \( \frac{1}{(x+1)(x^2+1)} \) as a sum of partial fractions. Set up the equation: \[ \frac{1}{(x+1)(x^2+1)} = \frac{A}{x+1} + \frac{Bx + C}{x^2 + 1} \] where \(A\), \(B\), and \(C\) are constants to be determined.
2. Multiply both sides of the equation by \( (x+1)(x^2+1) \) to clear the denominators, resulting in: \[ 1 = A(x^2 + 1) + (Bx + C)(x + 1) \]. Expand the right-hand side and collect like terms in powers of \(x\).
3. Equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations for \(A\), \(B\), and \(C\). Solve this system to find the values of these constants.
4. Rewrite the integral as the sum of integrals of the partial fractions: \[ \int_0^\infty \frac{dx}{(x+1)(x^2+1)} = \int_0^\infty \frac{A}{x+1} dx + \int_0^\infty \frac{Bx + C}{x^2 + 1} dx \].
5. Evaluate each integral separately. For \( \int \frac{1}{x+1} dx \), use the natural logarithm function. For \( \int \frac{x}{x^2 + 1} dx \), use substitution \( u = x^2 + 1 \). For \( \int \frac{1}{x^2 + 1} dx \), use the arctangent function. Then, apply the limits from 0 to \( \infty \) carefully to find the value of the original integral.

Key Concepts

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 bounds approach infinity or the points of discontinuity. This ensures the integral converges to a finite value.

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational function into simpler fractions that are easier to integrate. For example, a fraction with a quadratic and linear factor in the denominator can be expressed as a sum of simpler rational functions, facilitating straightforward integration.

Integration of Rational Functions

Integrating rational functions often involves recognizing standard integral forms and applying techniques like substitution or partial fractions. Understanding how to integrate terms like 1/(x² + 1) and 1/(x + a) is essential for solving integrals involving polynomial denominators.