Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problema 8.5.35

Capitolo 8, Problema 8.5.35

23-64. Integration Evaluate the following integrals.
35. ∫ (x² + 12x - 4)/(x³ - 4x) dx

Guida verificata passo dopo passo

Start by examining the integrand \( \frac{x^{2} + 12x - 4}{x^{3} - 4x} \). Notice that the denominator can be factored to simplify the expression. Factor \( x^{3} - 4x \) as \( x(x^{2} - 4) \), and further factor \( x^{2} - 4 \) as \( (x - 2)(x + 2) \). So the denominator becomes \( x(x - 2)(x + 2) \).

Rewrite the integral as \( \int \frac{x^{2} + 12x - 4}{x(x - 2)(x + 2)} \, dx \). Since the denominator is factored into linear terms, set up a partial fraction decomposition: \[ \frac{x^{2} + 12x - 4}{x(x - 2)(x + 2)} = \frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 2} \], where \( A, B, C \) are constants to be determined.

Multiply both sides of the equation by the denominator \( x(x - 2)(x + 2) \) to clear the fractions: \[ x^{2} + 12x - 4 = A(x - 2)(x + 2) + B x (x + 2) + C x (x - 2) \].

Expand the right-hand side and collect like terms in powers of \( x \). Then, equate the coefficients of \( x^{2} \), \( x \), and the constant term on both sides to form a system of equations for \( A, B, C \).

Solve the system of equations to find the values of \( A, B, C \). Once found, rewrite the integral as the sum of simpler integrals: \[ \int \frac{A}{x} \, dx + \int \frac{B}{x - 2} \, dx + \int \frac{C}{x + 2} \, dx \]. Each of these integrals can be evaluated using the natural logarithm function.

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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, typically linear or quadratic factors. This method is essential when integrating rational functions where the degree of the numerator is less than the degree of the denominator.

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Partial Fraction Decomposition: Distinct Linear Factors

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors, such as linear or quadratic terms. Factoring the denominator is crucial before applying partial fraction decomposition, as it reveals the simpler components needed for the decomposition. For example, factoring x³ - 4x as x(x² - 4) = x(x - 2)(x + 2) helps identify the terms in the denominator.

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Taylor Polynomials

Integration of Rational Functions

Integration of rational functions often requires rewriting the integrand into a form that can be integrated using standard formulas. After partial fraction decomposition, each simpler fraction corresponds to a known integral type, such as logarithmic or arctangent functions. Understanding these integral forms allows for the evaluation of the original integral step-by-step.

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Intro to Rational Functions

Pratica correlata

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59. Area of a segment of a circle

Use two approaches to show that the area of a cap (or segment) of a circle of radius r subtended by an angle θ (see figure) is given by:

A_seg = (1/2) r² (θ - sin θ)

b. Find the area using calculus.

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54–57. {Use of Tech} Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).

54. ∫(from 0 to π/2) sin⁶x dx = 5π/32

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92–98. Evaluate the following integrals.

94. ∫ (dt / (t³ + 1))

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77–86. Comparison Test Determine whether the following integrals converge or diverge.

84. ∫(from 1 to ∞) (2 + cos x) / x² dx

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23-64. Integration Evaluate the following integrals.