Textbook Question
82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.
84. ∫ (from 0 to π) sec²x dx*(Note: Potential improperness at x = π/2)*
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.R.71
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
71. ∫ (2x² - 4x)/(x² - 4) dx
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Start by examining the integral \(\int \frac{2x^{2} - 4x}{x^{2} - 4} \, dx\). Notice that the degree of the numerator and denominator are the same, so consider performing polynomial division first to simplify the integrand.
Perform polynomial division of the numerator \(2x^{2} - 4x\) by the denominator \(x^{2} - 4\). This will rewrite the integrand as a polynomial plus a proper rational function.
After the division, express the integrand as \(A + \frac{Bx + C}{x^{2} - 4}\), where \(A\), \(B\), and \(C\) are constants to be determined from the division.
Next, factor the denominator \(x^{2} - 4\) as \((x - 2)(x + 2)\) to prepare for partial fraction decomposition of the proper rational part \(\frac{Bx + C}{(x - 2)(x + 2)}\).
Set up the partial fraction decomposition: \(\frac{Bx + C}{(x - 2)(x + 2)} = \frac{D}{x - 2} + \frac{E}{x + 2}\), solve for constants \(D\) and \(E\), then integrate each term separately 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.
Partial Fraction Decomposition
Partial fraction decomposition breaks a rational function into simpler fractions that are easier to integrate. It is especially useful when the degree of the numerator is less than the degree of the denominator, allowing the integral to be expressed as a sum of simpler rational functions.
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Partial Fraction Decomposition: Distinct Linear Factors
Polynomial Division
Polynomial division is used when the degree of the numerator is equal to or greater than the degree of the denominator. It simplifies the integrand by dividing the polynomials, resulting in a polynomial plus a proper fraction, which can then be integrated using other techniques.
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07:00Taylor Polynomials
Basic Integration Techniques
Basic integration techniques include recognizing standard integral forms and applying substitution or direct integration rules. After simplifying the integrand, these methods help evaluate the integral of polynomials, rational functions, or simpler expressions obtained from decomposition.
Recommended video:
06:07Basic Rules for Definite Integrals
Related Practice
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
15. ∫ (from 1 to 2) (3x⁵ + 48x³ + 3x² + 16)/(x³ + 16x) dx
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Textbook Question
95–98. {Use of Tech} Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(n) for the given values of n.
96. ∫ (from 1 to 3) dx/(x³ + x + 1); n = 4
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
35. ∫ x³/√(4x² + 16) dx
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
54. ∫ dx/√(9x² - 25), x > 5/3
Textbook Question
95–98. {Use of Tech} Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(n) for the given values of n.
97. ∫ (from 0 to 1) tan(x²) dx; n = 40