2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
63. ∫ dx/(x² - 2x - 15)

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Start by recognizing that the integral involves a rational function with a quadratic denominator: \(\int \frac{dx}{x^{2} - 2x - 15}\). The first step is to factor the quadratic expression in the denominator.

Factor the quadratic \(x^{2} - 2x - 15\) by finding two numbers that multiply to \(-15\) and add to \(-2\). This gives \(x^{2} - 2x - 15 = (x - 5)(x + 3)\).

Rewrite the integral using the factored form: \(\int \frac{dx}{(x - 5)(x + 3)}\). This sets up the integral for partial fraction decomposition.

Set up the partial fraction decomposition: \(\frac{1}{(x - 5)(x + 3)} = \frac{A}{x - 5} + \frac{B}{x + 3}\), where \(A\) and \(B\) are constants to be determined.

Multiply both sides by \((x - 5)(x + 3)\) to clear denominators, resulting in \(1 = A(x + 3) + B(x - 5)\). Then, solve for \(A\) and \(B\) by substituting convenient values for \(x\) or by equating coefficients.

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Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a rational function into simpler fractions that are easier to integrate. It is especially useful when the denominator can be factored into linear or quadratic terms. For example, expressing 1/(x² - 2x - 15) as a sum of fractions with denominators (x - 5) and (x + 3) simplifies integration.

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials. For the integral ∫ dx/(x² - 2x - 15), factoring the denominator into (x - 5)(x + 3) is essential to apply partial fractions. Recognizing how to factor quadratics quickly aids in simplifying integrals.

Integration of Rational Functions

Integrating rational functions often requires rewriting the integrand into simpler parts, such as partial fractions. Once decomposed, each term can be integrated using basic formulas, like ∫ dx/(x - a) = ln|x - a| + C. Understanding these standard integrals helps solve complex rational integrals efficiently.

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.63. ∫ dx/(x² - 2x - 15)