Textbook Question
Solve the initial value problems in Exercises 67–70 for x as a function of t.
(3t⁴ + 4t² + 1) (dx/dt) = 2√3, x(1) = -π√3/4
1
views
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.5.4
Expand the quotients in Exercises 1–8 by partial fractions.
(2x + 2) / (x² - 2x + 1)
Verified step by step guidance1
Recognize that the denominator \(x^{2} - 2x + 1\) can be factored. Factor it as a perfect square: \(x^{2} - 2x + 1 = (x - 1)^{2}\).
Set up the partial fraction decomposition for the rational expression \(\frac{2x + 2}{(x - 1)^{2}}\). Since the denominator is a repeated linear factor, express it as \(\frac{A}{x - 1} + \frac{B}{(x - 1)^{2}}\).
Multiply both sides of the equation by the denominator \((x - 1)^{2}\) to clear the fractions: \(2x + 2 = A(x - 1) + B\).
Expand the right side: \(2x + 2 = A x - A + B\). Then, group like terms: \(2x + 2 = A x + (B - A)\).
Equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations: For \(x\), \(2 = A\); for the constant term, \(2 = B - A\). Solve this system to find \(A\) and \(B\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
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 express a rational function as a sum of simpler fractions, making integration or other operations easier. It involves breaking down a complex fraction into simpler terms with denominators that are factors of the original denominator.
Recommended video:
10:07
Partial Fraction Decomposition: Distinct Linear Factors
Factoring Quadratic Expressions
Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials or identifying perfect square trinomials. Recognizing the factorization of the denominator is essential for setting up the partial fractions correctly.
Recommended video:
13:42Partial Fraction Decomposition: Irreducible Quadratic Factors
Handling Repeated Factors in Denominators
When the denominator has repeated factors, such as (x - a)², partial fractions must include terms for each power of the repeated factor. This ensures the decomposition accounts for all components of the denominator for accurate expansion.
Recommended video:
11:26Partial Fraction Decomposition: Repeated Linear Factors
Related Practice
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x arctan(x) dx
Textbook Question
Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^x, and the line x = ln(2) about the line x = ln(2).
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫₁² (8 dx / (x² - 2x + 2))
Textbook Question
Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.
1
views
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ e√x / √x dx