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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.R.71
Capitolo 8, Problema 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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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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Taylor 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.
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Basic Rules for Definite Integrals
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Domanda del libro di testo

114. {Use of Tech} Arc length of the natural logarithm Consider the curve y = ln(x).

a. Find the length of the curve from x = 1 to x = a and call it L(a).

(Hint: The change of variables u = sqrt(x^2 + 1) allows evaluation by partial fractions.)

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Domanda del libro di testo

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

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Domanda del libro di testo

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

65. ∫ (from 0 to 1) dy/((y + 1)(y² + 1))

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Domanda del libro di testo

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

46. ∫ (x³ + 4x² + 12x + 4)/((x² + 4x + 10)²) dx

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Domanda del libro di testo

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

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Domanda del libro di testo

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.

82. ∫ (from -∞ to -1) dx/(x - 1)⁴

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