Textbook Question
2. When applying the formula for integration by parts, how do you choose the u and dv? How can you apply integration by parts to an integral of the form ∫ f(x) dx?
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.PE.12
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(x + 1) / (x² (x − 1))] dx
Verified step by step guidance1
Start by examining the integrand \( \frac{x + 1}{x^{2}(x - 1)} \). Notice that the denominator is a product of powers of \( x \) and \( (x - 1) \), suggesting that partial fraction decomposition could be a useful method.
Set up the partial fraction decomposition for \( \frac{x + 1}{x^{2}(x - 1)} \) as follows: \(\n\[\n\)\( \frac{x + 1}{x^{2}(x - 1)} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x - 1} \), \(\n\]\nwhere\) \( A \), \( B \), and \( C \) are constants to be determined.
Multiply both sides of the equation by the common denominator \( x^{2}(x - 1) \) to clear the fractions: \(\n\)\(\n\)\( x + 1 = A x (x - 1) + B (x - 1) + C x^{2} \).
Expand the right-hand side and collect like terms in powers of \( x \). Then, equate the coefficients of corresponding powers of \( x \) on both sides to form a system of equations for \( A \), \( B \), and \( C \).
Solve the system of equations to find the values of \( A \), \( B \), and \( C \). Once found, rewrite the integral as the sum of simpler integrals: \(\n\)\(\n\)\( \int \frac{A}{x} dx + \int \frac{B}{x^{2}} dx + \int \frac{C}{x - 1} dx \). Each of these can be integrated using basic integral formulas.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Substitution
Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function or a complicated expression.
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Substitution With an Extra Variable
Partial Fraction Decomposition
Partial fraction decomposition breaks down a complex rational function into simpler fractions that are easier to integrate. This method is essential when integrating rational functions where the degree of the numerator is less than the degree of the denominator. It allows the integral to be expressed as a sum of simpler terms.
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10:07Partial Fraction Decomposition: Distinct Linear Factors
Properties of Rational Functions
Understanding rational functions, which are ratios of polynomials, is crucial for integration. Recognizing factors in the denominator and numerator helps in choosing the right method, such as substitution or partial fractions. Analyzing the function's structure guides the simplification and integration process.
Recommended video:
06:21Properties of Functions
Related Practice
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ x·sec²x dx
Textbook Question
7. What is the goal of the method of partial fractions?
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
127. ∫ (ln x) / (x + x ln x) dx
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Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ x dx / √(2 − x)
Textbook Question
Evaluate the integrals in Exercises 37–44.
∫ tan³(x) sec³(x) dx