Use any method to evaluate the integrals in Exercises 55–66.
∫ (x + 2) / (x³ - 2x² - 3x) dx

Verified step by step guidance

1. Start by factoring the denominator \(x^{3} - 2x^{2} - 3x\). Factor out the common factor \(x\) first, then factor the quadratic expression that remains.
2. Rewrite the integral as \(\int \frac{x + 2}{x (x^{2} - 2x - 3)} \, dx\). Next, factor the quadratic \(x^{2} - 2x - 3\) into two binomials.
3. Express the integrand as a sum of partial fractions: \(\frac{x + 2}{x (x - 3)(x + 1)} = \frac{A}{x} + \frac{B}{x - 3} + \frac{C}{x + 1}\), where \(A\), \(B\), and \(C\) are constants to be determined.
4. Multiply both sides by the denominator \(x (x - 3)(x + 1)\) to clear the fractions, then equate coefficients of corresponding powers of \(x\) to form a system of equations for \(A\), \(B\), and \(C\).
5. Solve the system for \(A\), \(B\), and \(C\), then rewrite the integral as the sum of simpler integrals: \(\int \frac{A}{x} \, dx + \int \frac{B}{x - 3} \, dx + \int \frac{C}{x + 1} \, dx\). Integrate each term 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 factoring the denominator and expressing the integrand as a sum of simpler rational expressions. This method is especially useful when integrating rational functions where the degree of the numerator is less than the degree of the denominator.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors, such as linear or quadratic terms. This step is crucial before applying partial fraction decomposition because it reveals the denominator's roots and their multiplicities. For example, factoring the cubic denominator x³ - 2x² - 3x helps identify the terms needed for decomposition.

Integration of Rational Functions

Integration of rational functions often requires rewriting the integrand into simpler parts, such as partial fractions, to apply standard integral formulas. After decomposition, each term corresponds to a basic integral form, like ∫1/(x - a) dx or ∫1/(x² + bx + c) dx, which can be integrated using logarithmic or inverse trigonometric functions.