Textbook Question
Volume: Find the volume generated by revolving one arch of the curve y = sin x about the x-axis.
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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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.6.30
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ cos^(-1)(√x) / √x dx
Verified step by step guidance1
Identify the integral: \(\int \frac{\cos^{-1}(\sqrt{x})}{\sqrt{x}} \, dx\).
Choose a substitution to simplify the integral. Let \(u = \sqrt{x}\), which implies \(x = u^2\).
Differentiate \(x = u^2\) to find \(dx\): \(dx = 2u \, du\).
Rewrite the integral in terms of \(u\): replace \(\sqrt{x}\) with \(u\) and \(dx\) with \(2u \, du\). The integral becomes \(\int \frac{\cos^{-1}(u)}{u} \cdot 2u \, du\).
Simplify the integral expression: the \(u\) terms cancel, leaving \(2 \int \cos^{-1}(u) \, du\). Now, use integration techniques or a table of integrals to evaluate \(\int \cos^{-1}(u) \, du\).
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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 is a method used to simplify integrals by changing variables. It involves choosing a substitution that transforms the integral into a more familiar or standard form, making it easier to evaluate. This technique is especially useful when the integral contains composite functions.
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04:27
Substitution With an Extra Variable
Inverse Trigonometric Functions
Inverse trigonometric functions, like arccosine (cos⁻¹), are the inverses of the basic trigonometric functions. Understanding their properties and derivatives is essential for integrating expressions involving them. Recognizing how to handle these functions within integrals helps in applying substitution effectively.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Using Integral Tables
Integral tables provide standard forms of integrals and their solutions, which can save time in evaluation. After substitution simplifies the integral, matching it to a form in the table allows direct application of known results. Familiarity with common integral forms is key to using these tables efficiently.
Recommended video:
08:01Integration Using Partial Fractions
Related Practice
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 2 to ∞ of ((1 / ln x) dx)
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x² √(4x - 9))
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.
∫ (1 - x²)^(1/2) / x⁴ dx
Textbook Question
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ 1 / (x⁴ + x) dx
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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.
∫ dx / (x √(7 - x²))