Domanda del libro di testo
7–84. Evaluate the following integrals.
45. ∫ from 0 to ln 2 [1 / (1 + eˣ)²] dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 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)
Guida verificata passo dopo passo1
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.
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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.
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Percorso guidato
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.
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Percorso guidato
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.
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Percorso guidato
10:07Partial Fraction Decomposition: Distinct Linear Factors
Pratica correlata
Domanda del libro di testo
79. Tabular integration extended Refer to Exercise 77.
a. The following table shows the method of tabular integration applied to
∫ eˣ cos x dx.
Use the table to express ∫ eˣ cos x dx in terms of the sum of functions and an indefinite integral.
b. Solve the equation in part (a) for ∫ eʳ cos z dz.
c. Evaluate ∫ e⁻ᶻ sin 3z dz by applying the idea from parts (a) and (b).
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Domanda del libro di testo
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
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Domanda del libro di testo
6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.
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Domanda del libro di testo
23-64. Integration Evaluate the following integrals.
60.∫ 1/[(y² + 1)(y² + 2)] dy
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Domanda del libro di testo
Evaluate the following integrals.
∫ eˣ/(e²ˣ + 2eˣ + 17) dx
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