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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 8 - Integration Techniques문제 8.5.17
8장, 문제 8.5.17

17-22. Give the partial fraction decomposition for the following expressions.
17. (5x - 7) / (x² - 3x + 2)

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Start by factoring the denominator of the given rational expression. The denominator is \(x^{2} - 3x + 2\). Find two numbers that multiply to \(2\) and add to \(-3\).
Rewrite the denominator as a product of its factors: \(x^{2} - 3x + 2 = (x - 1)(x - 2)\).
Set up the partial fraction decomposition form for the expression \(\frac{5x - 7}{(x - 1)(x - 2)}\) as \(\frac{A}{x - 1} + \frac{B}{x - 2}\), where \(A\) and \(B\) are constants to be determined.
Multiply both sides of the equation by the denominator \((x - 1)(x - 2)\) to clear the fractions, resulting in \(5x - 7 = A(x - 2) + B(x - 1)\).
Expand the right side and collect like terms, then equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations. Solve this system to find the values of \(A\) and \(B\).

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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 express a rational function as a sum of simpler fractions with denominators that are factors of the original denominator. This method simplifies integration and other operations by breaking down complex fractions into manageable parts.
추천 영상:
10:07
Partial Fraction Decomposition: Distinct Linear Factors

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials. For example, x² - 3x + 2 factors into (x - 1)(x - 2). Factoring is essential in partial fraction decomposition to identify the denominators of the simpler fractions.
추천 영상:
13:42
Partial Fraction Decomposition: Irreducible Quadratic Factors

Setting Up and Solving Systems of Equations

After expressing the rational function as a sum of partial fractions, coefficients are determined by equating numerators and solving the resulting system of linear equations. This step ensures the decomposition accurately represents the original expression.
추천 영상:
5:02
Solving Logarithmic Equations
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교과서 질문

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7–84. Evaluate the following integrals.

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77–86. Comparison Test Determine whether the following integrals converge or diverge.

78. ∫(from 0 to ∞) dx / (eˣ + x + 1)

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60–69. Completing the square Evaluate the following integrals.

68. ∫ dx / sqrt((x - 1)(3 - x))

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