Textbook Question
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁰ x² e^(x³) 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 8.2.24
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ e^(-2x) sin(2x) dx
Verified step by step guidance1
Identify the integral to solve: \(\int e^{-2x} \sin(2x) \, dx\).
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Choose \(u\) and \(dv\) wisely. For this integral, let \(u = \sin(2x)\) and \(dv = e^{-2x} dx\).
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = 2 \cos(2x) \, dx\), and integrate \(dv\) to get \(v = \int e^{-2x} dx = -\frac{1}{2} e^{-2x}\).
Apply the integration by parts formula: substitute \(u\), \(v\), \(du\) into \(\int u \, dv = uv - \int v \, du\), then simplify the resulting integral. You may need to apply integration by parts a second time or use algebraic manipulation to solve for the original integral.
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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 based on the product rule for 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 is crucial to simplify the integral effectively.
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Integration by Parts for Definite Integrals
Integration of Exponential and Trigonometric Functions
Integrals involving products of exponential and trigonometric functions often require repeated application of integration by parts. Recognizing patterns and using algebraic manipulation helps to solve these integrals, sometimes leading to an equation involving the original integral.
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05:11Integrals of General Exponential Functions
Solving for the Original Integral
When integration by parts leads back to the original integral, you can solve for it algebraically. This involves setting up an equation where the integral appears on both sides and isolating it to find its value, a common strategy for integrals like ∫e^(ax) sin(bx) dx.
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5:59Graph Hyperbolas NOT at the Origin
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^π 3sin(x/3) dx
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Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫₀¹/√2 2x arcsin(x²) dx
Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ (x⁴ - 1) / (x⁵ - 5x + 1) dx
Textbook Question
In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x³ dx) / (x² - 2x + 1) from -1 to 0
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Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x² - 4) / x dx