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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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.44
Evaluate the integrals in Exercises 37–44.
∫ eᵗ √[tan²(eᵗ) + 1] dt
Verified step by step guidance1
Recognize that the integral is of the form \(\int e^{t} \sqrt{\tan^{2}(e^{t}) + 1} \, dt\). Notice that inside the square root, we have \(\tan^{2}(e^{t}) + 1\), which can be simplified using a trigonometric identity.
Recall the Pythagorean identity: \(1 + \tan^{2}(x) = \sec^{2}(x)\). Applying this, rewrite the integrand as \(e^{t} \sqrt{\sec^{2}(e^{t})}\).
Since \(\sqrt{\sec^{2}(e^{t})} = |\sec(e^{t})|\), and assuming the domain where \(\sec(e^{t})\) is positive, the integrand simplifies to \(e^{t} \sec(e^{t})\).
Use substitution to simplify the integral: let \(u = e^{t}\). Then, \(du = e^{t} dt\), which means \(du = e^{t} dt\) or equivalently \(e^{t} dt = du\). Substitute these into the integral to get \(\int \sec(u) \, du\).
Now, the integral reduces to \(\int \sec(u) \, du\), which is a standard integral. You can proceed by recalling the formula for \(\int \sec(u) \, du\) and then substitute back \(u = e^{t}\) to express the answer in terms of \(t\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Exponential Functions
Understanding how to integrate functions involving exponential terms like e^t is essential. This includes recognizing when substitution can simplify the integral, especially when the exponent appears inside another function.
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Integrals of General Exponential Functions
Trigonometric Identities
Familiarity with trigonometric identities, such as 1 + tan²(x) = sec²(x), helps simplify expressions under the integral. Applying these identities can transform complex radicals into more manageable forms.
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7:17Verifying Trig Equations as Identities
Substitution Method in Integration
The substitution method involves changing variables to simplify the integral. Identifying an inner function whose derivative appears elsewhere in the integrand allows rewriting the integral in a simpler form for direct integration.
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07:33Euler's Method
Related Practice
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