Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (x² √(x² + 1))
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 8.2.36
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (ln x)³/x dx
Verified step by step guidance1
Recognize that the integral is \( \int \frac{(\ln x)^3}{x} \, dx \). Notice that the integrand involves \( (\ln x)^3 \) divided by \( x \), which suggests a substitution related to \( \ln x \).
Use the substitution \( u = \ln x \). Then, the differential \( du = \frac{1}{x} dx \), which means \( dx = x \, du \). Since \( \frac{1}{x} dx = du \), the integral can be rewritten in terms of \( u \).
Rewrite the integral using the substitution: \( \int (\ln x)^3 \cdot \frac{1}{x} dx = \int u^3 \, du \). This simplifies the integral to a basic power function of \( u \).
Integrate \( u^3 \) with respect to \( u \) using the power rule for integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), where here \( n = 3 \).
After integrating, substitute back \( u = \ln x \) to express the answer in terms of \( x \). This completes the evaluation of the integral.
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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 simplifies integrals by changing variables to make the integral easier to solve. For example, setting u = ln(x) transforms the integral into a polynomial form in terms of u, which is easier to integrate.
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Substitution With an Extra Variable
Properties of Logarithmic Functions
Understanding the natural logarithm function ln(x) and its derivative is crucial. Since d/dx [ln(x)] = 1/x, this relationship helps identify substitution candidates and simplifies integrals involving ln(x).
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06:21Properties of Functions
Integration of Polynomial Functions
Once substitution reduces the integral to a polynomial form, integrating powers of a variable (like u³) involves applying the power rule for integration: ∫u^n du = u^(n+1)/(n+1) + C, which is straightforward and essential for solving the problem.
Recommended video:
07:00Taylor Polynomials
Related Practice
Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x - 2) / √(x - 1) dx
Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (x² - 1)^(5/2), where x > 1
Textbook Question
Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.
f(x) = (1/x) over [c, c + 1]
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² sin(x³) dx
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ dx / [(x + 1)(x² + 1)]
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