Chapter 7: Problem 7
Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{3 x-13}{x^{2}+3 x-10} d x\)
Short Answer
Expert verified
\( \ln|x+5| + 2\ln|x-2| + C \).
Step by step solution
01
Factor the Denominator
Notice that the denominator can be factored. The quadratic expression in the denominator is given by \( x^2 + 3x - 10 \). To factor this quadratic, we look for two numbers that multiply to -10 and add to 3. These numbers are 5 and -2, so we can write the denominator as \( (x + 5)(x - 2) \).
02
Set Up Partial Fractions
Now that the denominator is factored, express the integrand \( \frac{3x-13}{(x+5)(x-2)} \) as a sum of partial fractions: \( \frac{A}{x+5} + \frac{B}{x-2} \).
03
Solve for Coefficients
Multiply through by the common denominator \((x+5)(x-2)\) to clear the fractions: \[ 3x - 13 = A(x-2) + B(x+5) \].Expand and simplify: \[ 3x - 13 = Ax - 2A + Bx + 5B \].Combine like terms to get:\[ 3x - 13 = (A + B)x + (5B - 2A) \].By equating coefficients, we have the system of equations:\[ A + B = 3 \] \[ 5B - 2A = -13 \]. Solve this system to find \( A \) and \( B \). After calculations, \( A = 1 \) and \( B = 2 \).
04
Integrate Each Term
Substitute back \( A = 1 \) and \( B = 2 \) into the partial fractions to get: \( \frac{1}{x+5} + \frac{2}{x-2} \). The integral becomes:\[ \int \frac{1}{x+5} \, dx + \int \frac{2}{x-2} \, dx \].Integrate each term individually: \[ \ln|x+5| + 2\ln|x-2| + C \], where \( C \) is the constant of integration.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Techniques
Integration is a core concept in calculus, especially when dealing with complex expressions. One effective technique for integrating rational functions is the method of partial fraction decomposition. This method is particularly useful when handling integrands that are rational expressions, as it breaks them down into simpler fractions that are easier to integrate. The general steps involve factoring the polynomial in the denominator and then expressing the original fraction as a sum of simpler fractions. Each of these simpler fractions can then be integrated using basic integration rules.
In the given exercise, we started with the integrand \( \frac{3x-13}{x^2+3x-10} \). By factoring the denominator and setting up partial fractions, we simplified the process of integration. Once the expression is broken down into simpler fractions, integrating becomes a straightforward task. The end result is an integral that is much easier to handle and calculate.
To master these techniques, it's crucial to understand both the theory and the application. Practice with various rational expressions will deepen your understanding and enhance your skills.
In the given exercise, we started with the integrand \( \frac{3x-13}{x^2+3x-10} \). By factoring the denominator and setting up partial fractions, we simplified the process of integration. Once the expression is broken down into simpler fractions, integrating becomes a straightforward task. The end result is an integral that is much easier to handle and calculate.
To master these techniques, it's crucial to understand both the theory and the application. Practice with various rational expressions will deepen your understanding and enhance your skills.
Factoring Quadratics
Factoring is a crucial step in partial fraction decomposition, especially when the denominator is a quadratic expression. The first step is to express the quadratic in its factored form, which simplifies the integration process.
Quadratic expressions often take the form \( ax^2 + bx + c \). In our exercise, this was seen as \( x^2 + 3x - 10 \). To factor this, we look for two numbers that multiply to \( c \) (which is -10) and add to \( b \) (which is 3). In this instance, these numbers are 5 and -2, allowing us to express the quadratic as \((x + 5)(x - 2)\).
Understanding how to factor quadratics is essential not just for decomposition but also for other areas in math, like solving equations and simplifying expressions. Practicing different factoring techniques for various quadratic forms will improve your overall mathematical competence.
Quadratic expressions often take the form \( ax^2 + bx + c \). In our exercise, this was seen as \( x^2 + 3x - 10 \). To factor this, we look for two numbers that multiply to \( c \) (which is -10) and add to \( b \) (which is 3). In this instance, these numbers are 5 and -2, allowing us to express the quadratic as \((x + 5)(x - 2)\).
Understanding how to factor quadratics is essential not just for decomposition but also for other areas in math, like solving equations and simplifying expressions. Practicing different factoring techniques for various quadratic forms will improve your overall mathematical competence.
Rational Expressions
Rational expressions are fractions where both the numerator and the denominator are polynomials. The given problem involves the rational expression \( \frac{3x - 13}{x^2 + 3x - 10} \).
The goal with partial fraction decomposition is to express such rational expressions as a sum of simpler fractions with linear denominators where possible. This simplification enables the use of basic integration techniques, such as those derived from natural logarithms, to perform calculus operations effectively.
Handling rational expressions successfully requires comfort with polynomial operations, especially factoring and expanding. Once these polynomial manipulation skills are mastered, decomposing and integrating rational expressions become more intuitive and efficient.
The goal with partial fraction decomposition is to express such rational expressions as a sum of simpler fractions with linear denominators where possible. This simplification enables the use of basic integration techniques, such as those derived from natural logarithms, to perform calculus operations effectively.
Handling rational expressions successfully requires comfort with polynomial operations, especially factoring and expanding. Once these polynomial manipulation skills are mastered, decomposing and integrating rational expressions become more intuitive and efficient.
Calculus Integration Methods
Calculus offers a variety of integration methods, each suited for different types of functions and expressions. The method of partial fraction decomposition is just one of these, often used when dealing with rational expressions.
In our exercise, after using partial fraction decomposition, the integrals became \( \int \frac{1}{x+5} \, dx \) and \( \int \frac{2}{x-2} \, dx \). These can be easily integrated using the rule \( \int \frac{1}{u} \, du = \ln|u| + C \).
Other integration methods include substitution, integration by parts, and trigonometric substitution, each with its unique applications. Mastering these methods provides a comprehensive toolkit for tackling a wide range of integrals in calculus. Understanding when and how to apply each technique is crucial for solving complex problems efficiently.
In our exercise, after using partial fraction decomposition, the integrals became \( \int \frac{1}{x+5} \, dx \) and \( \int \frac{2}{x-2} \, dx \). These can be easily integrated using the rule \( \int \frac{1}{u} \, du = \ln|u| + C \).
Other integration methods include substitution, integration by parts, and trigonometric substitution, each with its unique applications. Mastering these methods provides a comprehensive toolkit for tackling a wide range of integrals in calculus. Understanding when and how to apply each technique is crucial for solving complex problems efficiently.
