23-64. Integration Evaluate the following integrals.
23. ∫ [3 / ((x - 1)(x + 2))] dx

Verified step by step guidance

1. Start by expressing the integrand \( \frac{3}{(x - 1)(x + 2)} \) as a sum of partial fractions. Assume it can be written as \( \frac{A}{x - 1} + \frac{B}{x + 2} \), where \(A\) and \(B\) are constants to be determined.
2. Multiply both sides of the equation by the denominator \( (x - 1)(x + 2) \) to clear the fractions, resulting in \( 3 = A(x + 2) + B(x - 1) \).
3. Expand the right-hand side to get \( 3 = A x + 2A + B x - B \), then group like terms: \( 3 = (A + B) x + (2A - B) \).
4. Set up a system of equations by equating the coefficients of corresponding powers of \(x\) on both sides. Since the left side has no \(x\) term, the coefficient of \(x\) must be zero, and the constant term must be 3. So, \( A + B = 0 \) and \( 2A - B = 3 \).
5. Solve the system for \(A\) and \(B\), then rewrite the integral as \( \int \left( \frac{A}{x - 1} + \frac{B}{x + 2} \right) dx \). Finally, integrate each term separately using the formula \( \int \frac{1}{x - c} dx = \ln|x - c| + C \).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a complex rational function into simpler fractions that are easier to integrate. It involves expressing the integrand as a sum of fractions with linear or quadratic denominators, allowing straightforward integration of each term.

Integration of Rational Functions

Integrating rational functions often requires rewriting the integrand into simpler parts, such as partial fractions. Once decomposed, each term can be integrated using basic integral formulas, typically involving logarithmic functions for linear denominators.

Logarithmic Integration

When integrating functions of the form 1/(ax + b), the result is a logarithmic function ln|ax + b|/a plus a constant. This concept is essential for integrating the terms obtained after partial fraction decomposition with linear denominators.