Textbook Question
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ from 0 to 1 x√(1 - x) 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 8.3.40
Evaluate the integrals in Exercises 33–52.
∫ eˣ sec³(eˣ) dx
Verified step by step guidance1
Recognize that the integral is of the form \(\int e^{x} \sec^{3}(e^{x}) \, dx\). Notice that the argument of the secant function is \(e^{x}\), which suggests a substitution involving \(e^{x}\).
Let \(u = e^{x}\). Then, compute the differential \(du = e^{x} dx\), which implies \(du = u \, dx\) or equivalently \(dx = \frac{du}{u}\).
Rewrite the integral in terms of \(u\): substitute \(e^{x} = u\) and \(dx = \frac{du}{u}\), so the integral becomes \(\int u \sec^{3}(u) \cdot \frac{du}{u} = \int \sec^{3}(u) \, du\).
Now, focus on evaluating \(\int \sec^{3}(u) \, du\). Recall that integrals of odd powers of secant can be handled by splitting \(\sec^{3}(u)\) as \(\sec(u) \cdot \sec^{2}(u)\) and using the identity \(\sec^{2}(u) = 1 + \tan^{2}(u)\).
Use the substitution \(t = \tan(u)\), so that \(dt = \sec^{2}(u) du\). Express the integral in terms of \(t\) and integrate accordingly, then substitute back to \(u\) and finally back to \(x\) to express the answer in terms of the original variable.
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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 involves changing variables to simplify an integral. It is useful when the integral contains a composite function, allowing you to rewrite the integral in terms of a new variable, making it easier to solve.
Recommended video:
04:27
Substitution With an Extra Variable
Trigonometric Identities for Secant
Understanding identities involving secant, such as expressing sec³(x) in terms of sec(x) and tan(x), helps in breaking down complex trigonometric integrals. These identities facilitate rewriting the integral into simpler parts that are easier to integrate.
Recommended video:
7:17Verifying Trig Equations as Identities
Integration of Powers of Secant
Integrating powers of secant functions often requires specific techniques, like using reduction formulas or rewriting powers in terms of sec and tan. Familiarity with these methods is essential to evaluate integrals involving sec³(x) or higher powers.
Recommended video:
05:42Example 6: Integral of Secant & Cosecant
Related Practice
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² dx / √|x − 1|
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Textbook Question
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.
∫₋₁¹ (√(1 + x²) sin x) dx
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Textbook Question
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from -1 to 1 of (dθ / (θ² - 2θ))
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Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^π sin⁵(x/2) dx
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Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ (x² - 2x + 1) e^(2x) dx