Textbook Question
Expand the quotients in Exercises 1–8 by partial fractions.
(5x - 7) / (x² - 3x + 2)
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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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.1.34
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.
∫ e^(z + eᶻ) dz
Verified step by step guidance1
Identify the integral to solve: \(\int e^{z + e^{z}} \, dz\).
Rewrite the integrand by separating the exponent: \(e^{z + e^{z}} = e^{z} \cdot e^{e^{z}}\).
Consider a substitution to simplify the integral. Let \(u = e^{z}\), so that \(\frac{du}{dz} = e^{z} = u\), which implies \(dz = \frac{du}{u}\).
Rewrite the integral in terms of \(u\): \(\int e^{z} \cdot e^{e^{z}} \, dz = \int u \cdot e^{u} \cdot \frac{du}{u} = \int e^{u} \, du\).
Integrate \(\int e^{u} \, du\) to get \(e^{u} + C\), then substitute back \(u = e^{z}\) to express the answer in terms of \(z\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Substitution
Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function.
Recommended video:
04:27
Substitution With an Extra Variable
Exponential Functions and Their Properties
Exponential functions have the form e^u, where u is a function of the variable. Understanding how to differentiate and integrate exponential functions, especially when the exponent is itself a function, is crucial. Recognizing the chain rule in reverse helps in integrating such expressions.
Recommended video:
06:21Properties of Functions
Recognizing Composite Functions in Integrals
Composite functions are functions within functions, such as e^(z + e^z). Identifying the inner function and its derivative within the integrand allows the use of substitution. This recognition simplifies the integral by reducing it to a basic form that is easier to evaluate.
Recommended video:
3:48Evaluate Composite Functions - Special Cases
Related Practice
Textbook Question
Average value
In a mass-spring-dashpot system like the one in Exercise 65, the mass's position at time t is
y = 4e^(-t)(sin(t) - cos(t)), t ≥ 0.
Find the average value of y over the interval 0 ≤ t ≤ 2π.
Textbook Question
Evaluate ∫ x³ √(1 - x²) dx using:
c. A trigonometric substitution.
Textbook Question
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Use numerical integration to estimate the value of
arcsin(0.6) = ∫ (from 0 to 0.6) dx / √(1 - x²).
For reference, arcsin(0.6) = 0.64350 to five decimal places.
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ tan^(-1)(x) / x² dx
Textbook Question
Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.
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