2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
71. ∫ (2x² - 4x)/(x² - 4) dx

Verified step by step guidance

1. Start by examining the integral \(\int \frac{2x^{2} - 4x}{x^{2} - 4} \, dx\). Notice that the degree of the numerator and denominator are the same, so consider performing polynomial division first to simplify the integrand.
2. Perform polynomial division of the numerator \$2x^{2} - 4x\( by the denominator \)x^{2} - 4$. This will rewrite the integrand as a polynomial plus a proper rational function.
3. After the division, express the integrand as \(A + \frac{Bx + C}{x^{2} - 4}\), where \(A\), \(B\), and \(C\) are constants to be determined from the division.
4. Next, factor the denominator \(x^{2} - 4\) as \((x - 2)(x + 2)\) to prepare for partial fraction decomposition of the proper rational part \(\frac{Bx + C}{(x - 2)(x + 2)}\).
5. Set up the partial fraction decomposition: \(\frac{Bx + C}{(x - 2)(x + 2)} = \frac{D}{x - 2} + \frac{E}{x + 2}\), solve for constants \(D\) and \(E\), then integrate each term separately using basic integral formulas.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition breaks a rational function into simpler fractions that are easier to integrate. It is especially useful when the degree of the numerator is less than the degree of the denominator, allowing the integral to be expressed as a sum of simpler rational functions.

Polynomial Division

Polynomial division is used when the degree of the numerator is equal to or greater than the degree of the denominator. It simplifies the integrand by dividing the polynomials, resulting in a polynomial plus a proper fraction, which can then be integrated using other techniques.

Basic Integration Techniques

Basic integration techniques include recognizing standard integral forms and applying substitution or direct integration rules. After simplifying the integrand, these methods help evaluate the integral of polynomials, rational functions, or simpler expressions obtained from decomposition.