Textbook Question
Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from −∞ to ∞ of (2 / (e^x + e^(−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.PE.2
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ x² ln(x) dx
Verified step by step guidance1
Identify the parts of the integral for integration by parts. Let \( u = \ln(x) \) because its derivative simplifies, and let \( dv = x^{2} dx \) because it is easy to integrate.
Compute the derivatives and integrals needed: \( du = \frac{1}{x} dx \) and \( v = \int x^{2} dx = \frac{x^{3}}{3} \).
Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the chosen \( u \), \( v \), \( du \) into the formula to get \( \int x^{2} \ln(x) dx = \ln(x) \cdot \frac{x^{3}}{3} - \int \frac{x^{3}}{3} \cdot \frac{1}{x} dx \).
Simplify the integral inside: \( \int \frac{x^{3}}{3} \cdot \frac{1}{x} dx = \frac{1}{3} \int x^{2} dx \).
Integrate \( \int x^{2} dx \) and write the full expression including the constant of integration \( C \).
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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 integration process.
Recommended video:
06:18
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 ln(x)), and dv is chosen as a function that is easy to integrate (like x² dx). This choice makes the integral manageable.
Recommended video:
07:51Choosing a Convergence Test
Logarithmic Functions in Integration
Integrals involving logarithmic functions often require integration by parts because ln(x) does not have a straightforward antiderivative. Differentiating ln(x) simplifies it to 1/x, which helps reduce the integral into a solvable form.
Recommended video:
5:26Graphs of Logarithmic Functions
Related Practice
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
133. ∫ (sin²x) / (1 + sin²x) dx
Textbook Question
A brief calculation shows that if 0 ≤ x ≤ 1, then the second derivative of
f(x) = √(1 + x⁴)
lies between 0 and 8.
Based on this, about how many subdivisions would you need to estimate the integral of f from 0 to 1
with an error no greater than 10⁻³ in absolute value using the Trapezoidal Rule?
Textbook Question
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(x³ + 1) / (x³ − x)] dx
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Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫₀³ (x + 2)√(x + 1) dx
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Textbook Question
Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to 3 of (1 / √(9 − x²)) dx
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