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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.GYR.7
Capitolo 8, Problema 8.GYR.7

7. What is the goal of the method of partial fractions?

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Understand that the method of partial fractions is used to decompose a complex rational function into a sum of simpler rational expressions whose denominators are factors of the original denominator.
Recognize that this decomposition makes it easier to perform operations such as integration or inverse Laplace transforms on the original rational function.
Identify the denominator of the given rational function and factor it completely into linear and/or irreducible quadratic factors.
Express the original rational function as a sum of fractions, each with one of the factors in the denominator and unknown constants in the numerators.
Solve for the unknown constants by multiplying both sides by the common denominator and equating coefficients or substituting convenient values of the variable.

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Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a complex rational function as a sum of simpler fractions. This simplification makes integration and other operations easier by breaking down complicated expressions into manageable parts.
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Partial Fraction Decomposition: Distinct Linear Factors

Rational Functions

A rational function is a ratio of two polynomials. Understanding the structure of rational functions is essential because partial fractions apply specifically to these types of functions, allowing us to rewrite them in simpler forms.
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Intro to Rational Functions

Integration of Rational Functions

One primary goal of partial fractions is to facilitate the integration of rational functions. By decomposing a complex fraction into simpler terms, each term can be integrated using basic integral formulas, making the overall integration process more straightforward.
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Intro to Rational Functions
Pratica correlata
Domanda del libro di testo

2. When applying the formula for integration by parts, how do you choose the u and dv? How can you apply integration by parts to an integral of the form ∫ f(x) dx?

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Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ x·sec²x dx

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Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

∫ dx / (1 + sin x + cos x)

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Finding volume

The region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.

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Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ θ·cos(2θ + 1) dθ

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Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [(x³ + 1) / (x³ − x)] dx

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