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 2 of (dx / (x ln x))
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 8.2.4
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ x² sin(x) dx
Verified step by step guidance1
Identify the parts of the integral for integration by parts. Let \( u = x^2 \) and \( dv = \sin(x) \, dx \).
Compute the derivatives and antiderivatives needed: \( du = 2x \, dx \) and \( v = -\cos(x) \) because the integral of \( \sin(x) \) is \( -\cos(x) \).
Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the expressions to get \( \int x^2 \sin(x) \, dx = -x^2 \cos(x) - \int -\cos(x) \cdot 2x \, dx \).
Simplify the integral: \( -x^2 \cos(x) + 2 \int x \cos(x) \, dx \). Now, you need to evaluate \( \int x \cos(x) \, dx \) using integration by parts again.
For the new integral \( \int x \cos(x) \, dx \), set \( u = x \) and \( dv = \cos(x) \, dx \), then repeat the integration by parts steps to solve it.
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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 integrating products like x² sin(x).
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Integration by Parts for Definite Integrals
Choosing u and dv
Selecting which part of the integrand to assign as u and which as dv is crucial. Typically, u is chosen as a function that simplifies when differentiated (like polynomials), and dv is chosen as a function that is easy to integrate (like trigonometric functions). This choice reduces the complexity of the resulting integral.
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07:51Choosing a Convergence Test
Repeated Application of Integration by Parts
When the integral involves powers of x multiplied by trigonometric functions, integration by parts may need to be applied multiple times. Each application reduces the power of x until the integral becomes straightforward. This iterative process is essential for solving integrals like ∫ x² sin(x) dx.
Recommended video:
08:23Repeated Integration by Parts
Related Practice
Textbook Question
Center of gravity: Find the center of gravity of the region bounded by the x-axis, the curve y = sec x, and the lines x = -pi/4 and x = pi/4.
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Textbook Question
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 3 sec^4(3x) dx
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / [x√(x² − 1)]
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Textbook Question
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to √3/2) dy / (1 - y²)^(5/2)
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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.
∫ (2 ln(z³)) / (16z) dz
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