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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.5.35
Chapter 8, Problem 8.5.35

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

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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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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, 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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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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Related Practice
Textbook Question

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.

Textbook Question

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

15. x / ((x⁴ - 16)²)

Textbook Question

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

Textbook Question

7–64. Integration review Evaluate the following integrals.

7. ∫ dx / (3 - 5x)^4

Textbook Question

23-64. Integration Evaluate the following integrals.

29. ∫₋₁² [(5x) / (x² - x - 6)] dx

Textbook Question

7–84. Evaluate the following integrals.

41. ∫ cot^(3/2)x · csc⁴x dx