Textbook Question
49. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample:
c. ∫ v du = u·v - ∫ u dv
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.5.3c
3. What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following?
c. A factor of (x² + 2x + 6) in the denominator
Verified step by step guidance1
Identify that the factor given, \(x^2 + 2x + 6\), is an irreducible quadratic factor because its discriminant \(\Delta = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 6 = 4 - 24 = -20 < 0\), which means it cannot be factored further over the real numbers.
Recall that for an irreducible quadratic factor \(ax^2 + bx + c\) in the denominator, the corresponding term in the partial fraction decomposition has the form \(\frac{Ax + B}{ax^2 + bx + c}\), where \(A\) and \(B\) are constants to be determined.
If the factor \(x^2 + 2x + 6\) appears with multiplicity 1 (i.e., it is not repeated), then the partial fraction term is simply \(\frac{Ax + B}{x^2 + 2x + 6}\).
If the factor appears with multiplicity greater than 1, say \(n\), then the decomposition includes terms for each power from 1 up to \(n\): \(\frac{A_1x + B_1}{x^2 + 2x + 6} + \frac{A_2x + B_2}{(x^2 + 2x + 6)^2} + \cdots + \frac{A_nx + B_n}{(x^2 + 2x + 6)^n}\).
Summarize that the key point is that each irreducible quadratic factor in the denominator corresponds to a numerator that is a linear polynomial \(Ax + B\) over that quadratic factor in the partial fraction decomposition.
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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 method used to express a proper rational function as a sum of simpler fractions whose denominators are factors of the original denominator. This technique simplifies integration and other operations by breaking complex fractions into manageable parts.
Recommended video:
10:07
Partial Fraction Decomposition: Distinct Linear Factors
Irreducible Quadratic Factors
An irreducible quadratic factor is a quadratic polynomial that cannot be factored further over the real numbers. In partial fraction decomposition, such factors in the denominator correspond to terms with linear numerators, for example, (Ax + B)/(quadratic factor).
Recommended video:
13:42Partial Fraction Decomposition: Irreducible Quadratic Factors
Proper Rational Function
A proper rational function is a fraction where the degree of the numerator is less than the degree of the denominator. This condition ensures that partial fraction decomposition can be applied directly without polynomial division.
Recommended video:
6:04Intro to Rational Functions
Related Practice
Textbook Question
45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.
45. ∫(0 to 1) e^(2x) dx; n = 25
c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.
Textbook Question
75. Exploring powers of sine and cosine
c. Prove that ∫₀ᵖⁱ sin²(nx) dx has the same value for all positive integers n.
Textbook Question
Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.
c.
π
∫ sin(mx) cos(nx) dx = 0, when |m + n| is even
0
Textbook Question
60. Two Methods
c. Verify that your answers to parts (a) and (b) are consistent.
Textbook Question
94. [Use of Tech] Skydiving A skydiver has a downward velocity given by v(t) = V_T [(1 - e^(-2gt/V_T))/(1 + e^(-2gt/V_T))],
where t = 0 is the instant the skydiver starts falling, g = 9.8 m/s² is the acceleration due to gravity, and V_T is the terminal velocity of the skydiver.
c. Verify by integration that the position function is given by
s(t) = V_T t + (V_T²/g) ln[(1 + e^(-2gt/V_T))/2],
where s'(t) = v(t) and s(0) = 0.
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