In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (2x + 1) / (x² - 7x + 12) dx

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First, factor the quadratic denominator \(x^{2} - 7x + 12\). Find two numbers that multiply to 12 and add to -7. This factors as \((x - 3)(x - 4)\).

Express the integrand as a sum of partial fractions: write \(\frac{2x + 1}{(x - 3)(x - 4)} = \frac{A}{x - 3} + \frac{B}{x - 4}\), where \(A\) and \(B\) are constants to be determined.

Multiply both sides of the equation by the denominator \((x - 3)(x - 4)\) to clear the fractions: \(2x + 1 = A(x - 4) + B(x - 3)\).

Expand the right side and collect like terms: \(2x + 1 = A x - 4A + B x - 3B = (A + B) x + (-4A - 3B)\). Then, equate the coefficients of \(x\) and the constant terms on both sides to form a system of equations: \(2 = A + B\) and \(1 = -4A - 3B\).

Solve the system of equations for \(A\) and \(B\). Once you find \(A\) and \(B\), rewrite the integral as \(\int \frac{A}{x - 3} dx + \int \frac{B}{x - 4} dx\) and integrate each term separately using the natural logarithm rule for integrals of the form \(\int \frac{1}{x - c} dx\).

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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 two binomials. This step is essential for partial fraction decomposition because it identifies the simpler linear factors in the denominator needed to split the integrand.

Integration of Rational Functions

Integration of rational functions often requires rewriting the integrand into simpler parts, such as partial fractions, to apply basic integration rules. Once decomposed, each term can be integrated using standard formulas, such as the integral of 1/(x - a) being ln|x - a| + C.

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.∫ (2x + 1) / (x² - 7x + 12) dx