In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x² + x) / (x⁴ - 3x² - 4) dx

Verified step by step guidance

1. First, factor the denominator \(x^{4} - 3x^{2} - 4\). Recognize it as a quadratic in terms of \(x^{2}\): let \(u = x^{2}\), so the expression becomes \(u^{2} - 3u - 4\). Factor this quadratic to get \((u - 4)(u + 1)\), which translates back to \((x^{2} - 4)(x^{2} + 1)\).
2. Next, factor \(x^{2} - 4\) further as a difference of squares: \((x - 2)(x + 2)\). So the full factorization of the denominator is \((x - 2)(x + 2)(x^{2} + 1)\).
3. Set up the partial fraction decomposition for the integrand \(\frac{x^{2} + x}{(x - 2)(x + 2)(x^{2} + 1)}\) as follows: \(\frac{x^{2} + x}{(x - 2)(x + 2)(x^{2} + 1)} = \frac{A}{x - 2} + \frac{B}{x + 2} + \frac{Cx + D}{x^{2} + 1}\), where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.
4. Multiply both sides of the equation by the denominator \((x - 2)(x + 2)(x^{2} + 1)\) to clear the fractions, resulting in: \(x^{2} + x = A(x + 2)(x^{2} + 1) + B(x - 2)(x^{2} + 1) + (Cx + D)(x - 2)(x + 2)\).
5. Expand the right-hand side and collect like terms in powers of \(x\). Then, equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations. Solve this system to find the values of \(A\), \(B\), \(C\), and \(D\). Once found, rewrite the integral as a sum of simpler integrals corresponding to each partial fraction term, which can then be integrated using standard techniques.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a complex rational function as a sum of simpler fractions. This method is especially useful when integrating rational functions, as it breaks down the integrand into terms that are easier to integrate individually.

Factoring Polynomials

Factoring the denominator polynomial is essential to apply partial fraction decomposition. By expressing the denominator as a product of linear or irreducible quadratic factors, we can set up the correct form for the partial fractions and solve for unknown coefficients.

Integration of Rational Functions

After decomposing the integrand into partial fractions, each term can be integrated using standard integral formulas. This often involves integrating terms like 1/(x - a), 1/(x² + bx + c), or polynomials, requiring knowledge of logarithmic and inverse trigonometric integrals.