Textbook Question
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [cos(θ) / (sin²(θ) + sin(θ) − 6)] dθ
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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.PE.28
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [1 / √(e^s + 1)] ds
Verified step by step guidance1
Identify the integral to solve: \(\int \frac{1}{\sqrt{e^{s} + 1}} \, ds\).
Consider a substitution to simplify the expression inside the square root. Let \(u = e^{s} + 1\).
Compute the differential \(du\): since \(u = e^{s} + 1\), then \(du = e^{s} \, ds\).
Rewrite \(ds\) in terms of \(du\) and \(s\): from \(du = e^{s} \, ds\), we get \(ds = \frac{du}{e^{s}}\). Also, note that \(e^{s} = u - 1\) from the substitution.
Substitute back into the integral to express it entirely in terms of \(u\): the integral becomes \(\int \frac{1}{\sqrt{u}} \cdot \frac{1}{u - 1} \, du\). Then proceed to simplify and integrate.
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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.
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Exponential Functions
Exponential functions have the form e^x, where e is Euler's number. Understanding their properties, such as their derivatives and integrals, is crucial since they often appear inside integrals. Recognizing how to handle e^s within an integral helps in applying substitution effectively.
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Integrals Involving Square Roots
Integrals containing square roots often require algebraic manipulation or substitution to simplify the root expression. Recognizing patterns like √(e^s + 1) helps in choosing an appropriate substitution to rewrite the integral in a more manageable form.
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07:01Integrals Involving Natural Logs: Substitution
Related Practice
Textbook Question
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Textbook Question
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Textbook Question
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(2x³ + x² − 21x + 24) / (x² + 2x − 8)] dx
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