Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ θ cos(πθ) dθ
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 8.5.26
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (s⁴ + 81) / (s(s² + 9)²) ds
Verified step by step guidance1
Identify the integrand: \( \frac{s^{4} + 81}{s (s^{2} + 9)^{2}} \). Since the denominator has a linear factor \( s \) and a repeated quadratic factor \( (s^{2} + 9)^{2} \), set up the partial fraction decomposition accordingly.
Write the partial fraction decomposition as: \[ \frac{s^{4} + 81}{s (s^{2} + 9)^{2}} = \frac{A}{s} + \frac{B s + C}{s^{2} + 9} + \frac{D s + E}{(s^{2} + 9)^{2}} \] where \( A, B, C, D, E \) are constants to be determined.
Multiply both sides of the equation by the denominator \( s (s^{2} + 9)^{2} \) to clear the fractions, resulting in: \[ s^{4} + 81 = A (s^{2} + 9)^{2} + (B s + C) s (s^{2} + 9) + (D s + E) s \].
Expand the right-hand side and collect like terms in powers of \( s \). Then, equate the coefficients of corresponding powers of \( s \) on both sides to form a system of equations for \( A, B, C, D, E \).
Solve the system of equations to find the values of \( A, B, C, D, E \). Once found, rewrite the integrand as the sum of partial fractions and integrate each term separately using standard integral formulas.
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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 simpler denominators, often linear or quadratic factors, raised to appropriate powers.
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Partial Fraction Decomposition: Distinct Linear Factors
Integration of Rational Functions
Integrating rational functions often requires rewriting the integrand into simpler terms via partial fractions. Once decomposed, each term can be integrated using standard formulas, such as logarithmic or inverse trigonometric integrals, depending on the denominator's form.
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Handling Repeated Quadratic Factors
When the denominator contains repeated quadratic factors, the partial fraction decomposition includes terms with increasing powers of the quadratic factor in the denominator. Each term typically has a linear numerator, and integrating these requires careful algebraic manipulation and knowledge of integration techniques for quadratic denominators.
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13:42Partial Fraction Decomposition: Irreducible Quadratic Factors
Related Practice
Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(1 - (ln x)²) / (x ln x) dx
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Textbook Question
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
85. Find the volume of the solid generated by revolving the region about the y-axis.
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Textbook Question
Expand the quotients in Exercises 1–8 by partial fractions.
z / (z³ - z² - 6z)
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Textbook Question
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
83. Find the area of the region.
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Textbook Question
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from π to ∞ of ((1 + sin x) / x² dx)
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