In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ dx / (x² + 2x)

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Start by factoring the denominator of the integrand. The denominator is \(x^{2} + 2x\), which can be factored as \(x(x + 2)\).

Set up the partial fraction decomposition for the integrand \(\frac{1}{x(x + 2)}\) as \(\frac{A}{x} + \frac{B}{x + 2}\), where \(A\) and \(B\) are constants to be determined.

Multiply both sides of the equation by the common denominator \(x(x + 2)\) to clear the fractions: \(1 = A(x + 2) + Bx\).

Expand and collect like terms: \(1 = A x + 2A + B x = (A + B) x + 2A\). Equate the coefficients of like powers of \(x\) on both sides to form a system of equations: for \(x\), \(A + B = 0\); for the constant term, \(2A = 1\).

Solve the system of equations to find the values of \(A\) and \(B\). Then rewrite the integral as the sum of two simpler integrals: \(\int \frac{A}{x} \, dx + \int \frac{B}{x + 2} \, dx\), which can be integrated using the natural logarithm function.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a complex rational function into simpler fractions that are easier to integrate. It involves expressing the integrand as a sum of fractions with simpler denominators, typically linear or irreducible quadratic factors.

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of simpler polynomials. For example, x² + 2x can be factored as x(x + 2), which is essential for setting up the partial fractions correctly.

Integration of Rational Functions

Integration of rational functions often requires rewriting the integrand into simpler terms using partial fractions. Once decomposed, each term can be integrated using basic integral formulas, such as ∫1/x dx = ln|x| + C.

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.∫ dx / (x² + 2x)