Textbook Question
85. Another form of ∫ sec x dx
a. Verify the identity:
sec x = cos x / (1 - sin² x)
b. Use the identity in part (a) to verify that:
∫ sec x dx = (1/2) ln |(1 + sin x)/(1 - sin x)| + C
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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 8.5.6
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.
6. (4x + 1)/(4x² - 1)
Verified step by step guidance1
Identify the denominator and factor it completely. The denominator is \(4x^{2} - 1\), which is a difference of squares and factors as \(\left(2x - 1\right)\left(2x + 1\right)\).
Since the denominator factors into two distinct linear factors, set up the partial fraction decomposition as a sum of fractions with unknown constants in the numerators over each linear factor:
\[\frac{4x + 1}{(2x - 1)(2x + 1)} = \frac{A}{2x - 1} + \frac{B}{2x + 1}\]
Here, \(A\) and \(B\) are constants to be determined later (but per the problem, we do not solve for them now).
This form is the appropriate setup for the partial fraction decomposition of the given expression.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas 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 a sum of fractions with simpler denominators, typically linear or quadratic factors.
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 other simpler polynomials. Recognizing how to factor the denominator, such as difference of squares, is essential for setting up the correct form of partial fractions.
Recommended video:
13:42Partial Fraction Decomposition: Irreducible Quadratic Factors
Form of Partial Fractions for Distinct Linear Factors
When the denominator factors into distinct linear terms, the partial fraction decomposition consists of separate fractions with unknown constants in the numerators over each linear factor. Setting up the correct form involves assigning constants to each linear factor without solving for them.
Recommended video:
10:07Partial Fraction Decomposition: Distinct Linear Factors
Related Practice
Textbook Question
Evaluate the following integrals.
∫ (1 - cosx)/(1 + cosx) dx
Textbook Question
6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.
Textbook Question
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
53. ∫ (from 0 to 1) ln x dx
1
views
Textbook Question
Evaluate the following integrals.
∫ eˣ/(e²ˣ + 2eˣ + 17) dx
Textbook Question
42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
42. The region bounded by f(x) = ln(x), y = 1, and the coordinate axes is revolved about the x-axis.