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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 8.6.85a
Chapter 8, Problem 8.6.85a

85. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. More than one integration method can be used to evaluate ∫ (1 / (1 - x²)) dx.

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Step 1: Recognize the integral ∫ (1 / (1 - x²)) dx and observe that the denominator (1 - x²) can be factored as (1 - x)(1 + x). This suggests that partial fraction decomposition might be a viable method to evaluate the integral.
Step 2: Recall that the integrand 1 / (1 - x²) resembles the derivative of the inverse hyperbolic tangent function, arctanh(x). This indicates that substitution involving inverse hyperbolic functions could also be used to evaluate the integral.
Step 3: Consider the possibility of trigonometric substitution. Since 1 - x² is a difference of squares, substituting x = sin(θ) or x = cos(θ) could simplify the integral into a trigonometric form that is easier to evaluate.
Step 4: Reflect on the fact that different integration methods often lead to the same result but may involve different intermediate steps. This is because integration methods are tools to manipulate the integrand into a form that can be integrated more easily.
Step 5: Conclude that more than one integration method can indeed be used to evaluate ∫ (1 / (1 - x²)) dx, and provide reasoning or examples for each method mentioned above to support the claim.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration Methods

Integration methods are techniques used to find the integral of a function. Common methods include substitution, integration by parts, and partial fractions. Each method has its own applicability depending on the form of the integrand, and sometimes multiple methods can yield the same result, providing flexibility in solving integrals.
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Euler's Method

Improper Integrals

The integral ∫ (1 / (1 - x²)) dx is an example of an improper integral, as it has vertical asymptotes at x = ±1. Understanding how to handle improper integrals is crucial, as they may require limits to evaluate the integral properly. This concept is essential for determining the convergence or divergence of the integral.
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Improper Integrals: Infinite Intervals

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving square roots or rational functions. For the integral ∫ (1 / (1 - x²)) dx, substituting x with sin(θ) or tan(θ) can transform the integrand into a more manageable form. This method highlights the versatility of integration techniques and the importance of recognizing when to apply them.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

65. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. To evaluate ∫ (4x⁶)/(x⁴ + 3x²) dx, the first step is to find the partial fraction decomposition of the integrand.

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Textbook Question

75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:

s(t) = e⁻ᵗ sin t

a. Graph the position function. At what times does the oscillator pass through the position s = 0?

Textbook Question

A piece of wood paneling must be cut in the shape shown in the figure.

The coordinates of several points on its curved surface are also shown (with units of inches).

a. Estimate the surface area of the paneling using the Trapezoid Rule.

Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

69. Let f(x) = sin(eˣ).

a. Find a Trapezoid Rule approximation to ∫[0 to 1] sin(eˣ) dx using n = 40 subintervals.

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Textbook Question

42. Approximating integrals The function f is twice differentiable on (-∞, ∞). Values of f at various points on [0, 20] are given in the table.

a. Approximate ∫(0 to 120) f(x) dx in three way using a left Riemann sum, a right Riemann sum and the Trapezoid Rule

Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

67. Let f(x) = √(x³ + 1).

a. Find a Midpoint Rule approximation to ∫[1 to 6] √(x³ + 1) dx using n = 50 subintervals.