Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x - 2) / √(x - 1) 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 8.2.40
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² sin(x³) dx
Verified step by step guidance1
Identify the integral to solve: \(\int x^{2} \sin(x^{3}) \, dx\).
Look for a substitution that simplifies the integral. Notice that the argument of the sine function is \(x^{3}\), and the derivative of \(x^{3}\) is \$3x^{2}\(, which is similar to the \)x^{2}$ term outside the sine.
Set the substitution: let \(u = x^{3}\). Then, compute the differential: \(du = 3x^{2} \, dx\), which implies \(x^{2} \, dx = \frac{du}{3}\).
Rewrite the integral in terms of \(u\): \(\int x^{2} \sin(x^{3}) \, dx = \int \sin(u) \cdot \frac{du}{3} = \frac{1}{3} \int \sin(u) \, du\).
Integrate \(\sin(u)\) with respect to \(u\): \(\int \sin(u) \, du = -\cos(u) + C\). Then substitute back \(u = x^{3}\) to express the answer in terms of \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. For example, setting u = x³ converts the integral into one involving sin(u), making it easier to integrate.
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Euler's Method
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation, used to integrate products of functions. It is applied when substitution is not straightforward, but in this problem, it may not be necessary.
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06:18Integration by Parts for Definite Integrals
Recognizing When to Use Each Technique
Identifying the most efficient integration method is crucial. Here, recognizing that the integral can be solved by substitution rather than integration by parts saves time and simplifies the process.
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05:03Initial Value Problems
Related Practice
Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ (x + 2) / (x³ - 2x² - 3x) 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
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (ln x)³/x dx
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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
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ dx / [(x + 1)(x² + 1)]
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