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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 8.5.65a
Chapter 8, Problem 8.5.65a

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.

Verified step by step guidance
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First, examine the integrand \( \frac{4x^{6}}{x^{4} + 3x^{2}} \). Notice that both numerator and denominator are polynomials in terms of \( x \).
Check the degree of the numerator and denominator: the numerator has degree 6, and the denominator has degree 4. Since the degree of the numerator is higher, consider simplifying the integrand by polynomial division before attempting partial fraction decomposition.
Perform polynomial division of \( 4x^{6} \) by \( x^{4} + 3x^{2} \) to rewrite the integrand as a polynomial plus a proper rational function (where the numerator degree is less than the denominator degree).
After polynomial division, if the remaining rational function has a denominator that can be factored into linear or irreducible quadratic factors, then partial fraction decomposition can be applied to that part.
Therefore, the first step is not to directly find the partial fraction decomposition of the original integrand, but to simplify it first by polynomial division. This shows that the statement is false.

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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 rational functions into simpler fractions that are easier to integrate. It applies when the integrand is a ratio of polynomials where the degree of the numerator is less than the degree of the denominator. This method is not suitable if the numerator's degree is equal to or greater than the denominator's.
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Partial Fraction Decomposition: Distinct Linear Factors

Polynomial Degree Comparison

Before applying partial fraction decomposition, compare the degrees of the numerator and denominator polynomials. If the numerator's degree is higher or equal, perform polynomial division first to rewrite the integrand as a polynomial plus a proper fraction. This step ensures the integrand fits the form required for partial fractions.
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Taylor Polynomials

Integration of Rational Functions

Integrating rational functions often involves simplifying the integrand through algebraic manipulation such as polynomial division or partial fractions. Recognizing the appropriate method depends on the form of the integrand, which guides the integration strategy and helps avoid unnecessary or incorrect steps.
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Intro to Rational Functions
Related Practice
Textbook Question

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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Textbook Question

Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.

a. Find the area of the region bounded by the graph of f and the x-axis on the interval [0,2].

Textbook Question

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

a. If m is a positive integer, then ∫[0 to π] cos^(2m+1)(x) dx = 0.

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

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

81. Possible and impossible integrals

Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.

a. I₀ = ∫ e⁻ˣ² dx cannot be expressed in terms of elementary functions. Evaluate I₁.