Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 8.5.14
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (y + 4) / (y² + y) dy from 1/2 to 1
Verified step by step guidance1
Start by factoring the denominator of the integrand. The denominator is \(y^{2} + y\), which can be factored as \(y(y + 1)\).
Express the integrand \(\frac{y + 4}{y(y + 1)}\) as a sum of partial fractions in the form \(\frac{A}{y} + \frac{B}{y + 1}\), where \(A\) and \(B\) are constants to be determined.
Multiply both sides of the equation by the common denominator \(y(y + 1)\) to clear the fractions: \(y + 4 = A(y + 1) + By\).
Expand the right side and collect like terms: \(y + 4 = Ay + A + By = (A + B)y + A\). Then, equate the coefficients of like terms from both sides to form a system of equations: for \(y\) terms, \(1 = A + B\); for constant terms, \(4 = A\).
Solve the system for \(A\) and \(B\), substitute these values back into the partial fractions, and then integrate each term separately over the interval from \(\frac{1}{2}\) to \(1\).
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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 technique used to break down a complex rational function into simpler fractions that are easier to integrate. It involves expressing the integrand as a sum of fractions with simpler denominators, typically linear or irreducible quadratic factors.
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Partial Fraction Decomposition: Distinct Linear Factors
Integration of Rational Functions
Integrating rational functions often requires rewriting the integrand into simpler parts, such as partial fractions. Once decomposed, each term can be integrated using basic integral formulas, including logarithmic and inverse trigonometric functions depending on the denominator.
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6:04Intro to Rational Functions
Definite Integrals and Limits of Integration
A definite integral calculates the net area under a curve between two points. After finding the antiderivative, the Fundamental Theorem of Calculus is applied by evaluating the antiderivative at the upper and lower limits and subtracting to find the integral's value.
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05:43Definition of the Definite Integral
Related Practice
Textbook Question
Solve the initial value problems in Exercises 53–56 for y as a function of x.
(x² + 1)² (dy/dx) = √(x² + 1), where y(0) = 1
Textbook Question
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 8 cot^4(t) dt
Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (4 + x²)
Textbook Question
87. Find the area of the region that lies between the curves y = sec x and y = tan x from x = 0 to x = π/2.
Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / √(1 - x²)