23-64. Integration Evaluate the following integrals.
32. ∫ (4x - 2)/(x³ - x) dx

Verified step by step guidance

1. First, observe the integral \( \int \frac{4x - 2}{x^{3} - x} \, dx \). Notice that the denominator can be factored. Factor \( x^{3} - x \) as \( x(x^{2} - 1) \), which further factors to \( x(x - 1)(x + 1) \).
2. Rewrite the integral using partial fraction decomposition. Express \( \frac{4x - 2}{x(x - 1)(x + 1)} \) as \( \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 1} \), where \( A, B, C \) are constants to be determined.
3. Multiply both sides of the equation by the denominator \( x(x - 1)(x + 1) \) to clear the fractions, resulting in an equation involving \( A, B, C \) and powers of \( x \). Equate the coefficients of corresponding powers of \( x \) on both sides to form a system of linear equations.
4. Solve the system of equations to find the values of \( A, B, C \). Once these constants are found, rewrite the integral as the sum of simpler integrals: \( \int \frac{A}{x} \, dx + \int \frac{B}{x - 1} \, dx + \int \frac{C}{x + 1} \, dx \).
5. Integrate each term separately using the formula \( \int \frac{1}{x - a} \, dx = \ln|x - a| + C \). Combine the results and include the constant of integration \( C \) to write the final expression for the integral.

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 simpler denominators, typically linear or quadratic factors. This method is essential when integrating rational functions where the degree of the numerator is less than the degree of the denominator.

Integration of Rational Functions

Integrating rational functions often requires rewriting the integrand into a form suitable for direct integration, such as sums of simpler fractions. After decomposition, each term can be integrated using basic integral formulas, including logarithmic and inverse trigonometric functions. Understanding how to handle these integrals is key to solving problems involving rational expressions.

Factoring Polynomials

Factoring polynomials is the process of expressing a polynomial as a product of its factors, which simplifies the integrand and aids in partial fraction decomposition. Recognizing common factors or special products (like difference of squares) helps break down the denominator into linear or quadratic factors. This step is crucial for setting up the integral correctly.