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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.R.63
Capitolo 8, Problema 8.R.63

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
63. ∫ dx/(x² - 2x - 15)

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Start by recognizing that the integral involves a rational function with a quadratic denominator: \(\int \frac{dx}{x^{2} - 2x - 15}\). The first step is to factor the quadratic expression in the denominator.
Factor the quadratic \(x^{2} - 2x - 15\) by finding two numbers that multiply to \(-15\) and add to \(-2\). This gives \(x^{2} - 2x - 15 = (x - 5)(x + 3)\).
Rewrite the integral using the factored form: \(\int \frac{dx}{(x - 5)(x + 3)}\). This sets up the integral for partial fraction decomposition.
Set up the partial fraction decomposition: \(\frac{1}{(x - 5)(x + 3)} = \frac{A}{x - 5} + \frac{B}{x + 3}\), where \(A\) and \(B\) are constants to be determined.
Multiply both sides by \((x - 5)(x + 3)\) to clear denominators, resulting in \(1 = A(x + 3) + B(x - 5)\). Then, solve for \(A\) and \(B\) by substituting convenient values for \(x\) or by equating coefficients.

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Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a rational function into simpler fractions that are easier to integrate. It is especially useful when the denominator can be factored into linear or quadratic terms. For example, expressing 1/(x² - 2x - 15) as a sum of fractions with denominators (x - 5) and (x + 3) simplifies integration.
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Partial Fraction Decomposition: Distinct Linear Factors

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials. For the integral ∫ dx/(x² - 2x - 15), factoring the denominator into (x - 5)(x + 3) is essential to apply partial fractions. Recognizing how to factor quadratics quickly aids in simplifying integrals.
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Partial Fraction Decomposition: Irreducible Quadratic Factors

Integration of Rational Functions

Integrating rational functions often requires rewriting the integrand into simpler parts, such as partial fractions. Once decomposed, each term can be integrated using basic formulas, like ∫ dx/(x - a) = ln|x - a| + C. Understanding these standard integrals helps solve complex rational integrals efficiently.
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Intro to Rational Functions
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Domanda del libro di testo

122. Comparing areas The region R₁ is bounded by the graph of y = tan(x) and the x-axis on the interval [0, π/3].

The region R₂ is bounded by the graph of y = sec(x) and the x-axis on the interval [0, π/6]. Which region has the greater area?

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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.

22. ∫ tan³ 5θ dθ

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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.

6. ∫ (2 − sin 2θ)/cos² 2θ dθ

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Domanda del libro di testo

118. Two worthy integrals

b. Let f be any positive continuous function on the interval [0, π/2]. Evaluate

∫ from 0 to π/2 of [f(cos x) / (f(cos x) + f(sin x))] dx.

(Hint: Use the identity cos(π/2 − x) = sin x.)


(Source: Mathematics Magazine 81, 2, Apr 2008)

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Domanda del libro di testo

120. Equal volumes

a. Let R be the region bounded by the graph of f(x) = x^(-p) and the x-axis, for x ≥ 1. Let V₁ and V₂ be the volumes of the solids generated when R is revolved about the x-axis and the y-axis, respectively, if they exist. For what values of p (if any) is V₁ = V₂?

b. Repeat part (a) on the interval [0, 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.

32. ∫ csc²(6x) cot(6x) dx

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