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.
123. ∫ √x * √(1 + √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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.PE.18
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(3v − 7) / ((v − 1)(v − 2)(v − 3))] dv
Verified step by step guidance1
Identify the integral to be solved: \(\int \frac{3v - 7}{(v - 1)(v - 2)(v - 3)} \, dv\).
Recognize that the integrand is a rational function where the degree of the numerator is less than the degree of the denominator, so partial fraction decomposition is appropriate.
Set up the partial fraction decomposition: express \(\frac{3v - 7}{(v - 1)(v - 2)(v - 3)}\) as \(\frac{A}{v - 1} + \frac{B}{v - 2} + \frac{C}{v - 3}\), where \(A\), \(B\), and \(C\) are constants to be determined.
Multiply both sides of the equation by the denominator \((v - 1)(v - 2)(v - 3)\) to clear the fractions, resulting in an equation involving \(3v - 7\) and the linear combination of \(A(v - 2)(v - 3)\), \(B(v - 1)(v - 3)\), and \(C(v - 1)(v - 2)\).
Solve for the constants \(A\), \(B\), and \(C\) by substituting convenient values for \(v\) (such as \(v=1\), \(v=2\), and \(v=3\)) or by equating coefficients, then integrate each resulting simpler fraction separately.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 linear or quadratic denominators, which can then be integrated individually.
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Partial Fraction Decomposition: Distinct Linear Factors
Integration of Rational Functions
Integrating rational functions often requires rewriting the integrand into a simpler form, such as partial fractions. Once decomposed, each term corresponds to a standard integral form, allowing the use of basic integration rules like logarithmic or arctangent integrals.
Recommended video:
6:04Intro to Rational Functions
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It is especially useful when the integrand contains composite functions or when the derivative of a function inside the integral appears elsewhere in the integrand.
Recommended video:
07:33Euler's Method
Related Practice
Textbook Question
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [t / √(4t² − 1)] dt
Textbook Question
Evaluate the integrals in Exercises 37–44.
∫ cos⁵(x) sin⁵(x) dx
Textbook Question
Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from 1 to ∞ of ((ln z) / z) dz
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.
∫ (e^x + e^(3x)) / e^(2x) dx
Textbook Question
Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from 6 to ∞ of (1 / √(θ² + 1)) dθ
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