7–84. Evaluate the following integrals.
16. ∫ [1 / (x⁴ – 1)] dx

Verified step by step guidance

1. Start by recognizing that the integrand is a rational function: \(\frac{1}{x^{4} - 1}\). The first step is to factor the denominator \(x^{4} - 1\) using the difference of squares formula: \(a^{2} - b^{2} = (a - b)(a + b)\). Here, \(x^{4} - 1 = (x^{2})^{2} - 1^{2} = (x^{2} - 1)(x^{2} + 1)\).
2. Next, factor \(x^{2} - 1\) further since it is also a difference of squares: \(x^{2} - 1 = (x - 1)(x + 1)\). So the full factorization of the denominator is \(x^{4} - 1 = (x - 1)(x + 1)(x^{2} + 1)\).
3. Rewrite the integral as \(\int \frac{1}{(x - 1)(x + 1)(x^{2} + 1)} \, dx\). The next step is to decompose the integrand into partial fractions of the form: \(\frac{A}{x - 1} + \frac{B}{x + 1} + \frac{Cx + D}{x^{2} + 1}\), where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.
4. Multiply both sides of the equation by the denominator \((x - 1)(x + 1)(x^{2} + 1)\) to clear the fractions, resulting in an identity involving polynomials. Then, equate the coefficients of corresponding powers of \(x\) on both sides to form a system of linear equations for \(A\), \(B\), \(C\), and \(D\).
5. Solve the system of equations to find the values of \(A\), \(B\), \(C\), and \(D\). Once these constants are found, rewrite the integral as a sum of simpler integrals: \(\int \frac{A}{x - 1} \, dx + \int \frac{B}{x + 1} \, dx + \int \frac{Cx + D}{x^{2} + 1} \, dx\). Each of these integrals can then be evaluated using standard integration techniques.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational functions into simpler fractions that are easier to integrate. For integrals involving polynomials in the denominator, factoring the denominator and expressing the integrand as a sum of simpler rational expressions allows straightforward integration.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. For the integral ∫1/(x⁴ - 1) dx, recognizing that x⁴ - 1 is a difference of squares helps factor it into (x² - 1)(x² + 1), and further factoring x² - 1 into (x - 1)(x + 1) simplifies the integrand for partial fractions.

Integration of Rational Functions

Integrating rational functions often requires rewriting the integrand into simpler terms using algebraic techniques like partial fractions. Once decomposed, each term corresponds to standard integral forms, such as logarithmic or arctangent functions, which can be integrated using known formulas.