In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x² dx) / ((x - 1)(x² + 2x + 1))

Verified step by step guidance

1. First, recognize that the denominator can be factored further. Notice that \(x^{2} + 2x + 1\) is a perfect square trinomial, so rewrite it as \((x + 1)^{2}\). Thus, the integral becomes \(\int \frac{x^{2}}{(x - 1)(x + 1)^{2}} \, dx\).
2. Set up the partial fraction decomposition for the integrand. Since the denominator has a linear factor \((x - 1)\) and a repeated linear factor \((x + 1)^{2}\), express the fraction as: \(\frac{x^{2}}{(x - 1)(x + 1)^{2}} = \frac{A}{x - 1} + \frac{B}{x + 1} + \frac{C}{(x + 1)^{2}}\).
3. Multiply both sides of the equation by the denominator \((x - 1)(x + 1)^{2}\) to clear the fractions, resulting in: \(x^{2} = A(x + 1)^{2} + B(x - 1)(x + 1) + C(x - 1)\).
4. Expand the right-hand side and collect like terms in powers of \(x\). This will give you a polynomial equation in \(x\) where the coefficients of corresponding powers of \(x\) on both sides must be equal. Use this to set up a system of equations to solve for \(A\), \(B\), and \(C\).
5. Once you find the values of \(A\), \(B\), and \(C\), rewrite the integral as the sum of simpler integrals: \(\int \frac{A}{x - 1} \, dx + \int \frac{B}{x + 1} \, dx + \int \frac{C}{(x + 1)^{2}} \, dx\). Then, integrate each term separately using standard integral formulas.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is especially useful when integrating rational functions, as it breaks down complex expressions into manageable parts that can be integrated individually.

Integration of Rational Functions

Integrating rational functions often involves rewriting the integrand into simpler terms, such as partial fractions. Once decomposed, each term can be integrated using standard formulas, including logarithmic and arctangent integrals depending on the denominator's form.

Handling Repeated and Quadratic Factors

When the denominator contains repeated linear factors or irreducible quadratic factors, the partial fraction decomposition includes terms with powers of linear factors and linear numerators over quadratics. Recognizing and correctly setting up these terms is essential for accurate integration.