23-64. Integration Evaluate the following integrals.
35. ∫ (x² + 12x - 4)/(x³ - 4x) dx

Verified step by step guidance

1. Start by examining the integrand \( \frac{x^{2} + 12x - 4}{x^{3} - 4x} \). Notice that the denominator can be factored to simplify the expression. Factor \( x^{3} - 4x \) as \( x(x^{2} - 4) \), and further factor \( x^{2} - 4 \) as \( (x - 2)(x + 2) \). So the denominator becomes \( x(x - 2)(x + 2) \).
2. Rewrite the integral as \( \int \frac{x^{2} + 12x - 4}{x(x - 2)(x + 2)} \, dx \). Since the denominator is factored into linear terms, set up a partial fraction decomposition: \[ \frac{x^{2} + 12x - 4}{x(x - 2)(x + 2)} = \frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 2} \], where \( A, B, C \) are constants to be determined.
3. Multiply both sides of the equation by the denominator \( x(x - 2)(x + 2) \) to clear the fractions: \[ x^{2} + 12x - 4 = A(x - 2)(x + 2) + B x (x + 2) + C x (x - 2) \].
4. Expand the right-hand side and collect like terms in powers of \( x \). Then, equate the coefficients of \( x^{2} \), \( x \), and the constant term on both sides to form a system of equations for \( A, B, C \).
5. Solve the system of equations to find the values of \( A, B, C \). Once found, rewrite the integral as the sum of simpler integrals: \[ \int \frac{A}{x} \, dx + \int \frac{B}{x - 2} \, dx + \int \frac{C}{x + 2} \, dx \]. Each of these integrals can be evaluated using the natural logarithm function.

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.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors, such as linear or quadratic terms. Factoring the denominator is crucial before applying partial fraction decomposition, as it reveals the simpler components needed for the decomposition. For example, factoring x³ - 4x as x(x² - 4) = x(x - 2)(x + 2) helps identify the terms in the denominator.

Integration of Rational Functions

Integration of rational functions often requires rewriting the integrand into a form that can be integrated using standard formulas. After partial fraction decomposition, each simpler fraction corresponds to a known integral type, such as logarithmic or arctangent functions. Understanding these integral forms allows for the evaluation of the original integral step-by-step.