Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ eˣ sec³(eˣ) dx
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 8.2.10
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ (x² - 2x + 1) e^(2x) dx
Verified step by step guidance1
Identify the integral to solve: \(\int (x^{2} - 2x + 1) e^{2x} \, dx\).
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Choose \(u\) and \(dv\) wisely. Let \(u = (x^{2} - 2x + 1)\) because its derivative simplifies the polynomial, and let \(dv = e^{2x} \, dx\) because the integral of an exponential function is straightforward.
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = (2x - 2) \, dx\), and integrate \(dv\) to get \(v = \frac{1}{2} e^{2x}\).
Apply the integration by parts formula: write \(\int (x^{2} - 2x + 1) e^{2x} \, dx = (x^{2} - 2x + 1) \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} (2x - 2) \, dx\), then simplify and prepare to integrate the remaining integral, which may require applying integration by parts again.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Parts
Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely simplifies the problem, especially when one function becomes simpler upon differentiation.
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Integration by Parts for Definite Integrals
Polynomial and Exponential Functions
Understanding the behavior of polynomial functions like x² - 2x + 1 and exponential functions like e^(2x) is crucial. Polynomials simplify upon differentiation, while exponentials remain similar upon integration and differentiation, making them suitable candidates for integration by parts.
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6:13Exponential Functions
Repeated Application of Integration by Parts
When integrating products involving polynomials and exponentials, multiple iterations of integration by parts may be necessary. Each step reduces the polynomial degree, eventually leading to an integral that can be directly evaluated, ensuring the problem is fully solved.
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08:23Repeated Integration by Parts
Related Practice
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
In Exercises 33–38, perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral.
∫ 2y⁴ / (y³ - y² + y - 1) dy
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Textbook Question
Solve the initial value problems in Exercises 67–70 for x as a function of t.
(t + 1) (dx/dt) = x² + 1 (for t > -1), x(0) = 0
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀⁴ dx / √(4 − x)
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