Textbook Question
Evaluate the integrals in Exercises 23–32.
∫₀^π √(1 - cos(2x)) 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 8.6.8
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x² √(4x - 9))
Verified step by step guidance1
Identify the integral to solve: \(\int \frac{dx}{x^{2} \sqrt{4x - 9}}\).
Look for a suitable substitution to simplify the square root expression. Since the integrand contains \(\sqrt{4x - 9}\), consider the substitution \(t = \sqrt{4x - 9}\) or express \(x\) in terms of \(t\) to simplify the root.
Rewrite \(x\) and \(dx\) in terms of \(t\) using the substitution. For example, if \(t = \sqrt{4x - 9}\), then \(t^{2} = 4x - 9\), which implies \(x = \frac{t^{2} + 9}{4}\). Differentiate to find \(dx\) in terms of \(dt\).
Substitute \(x\) and \(dx\) back into the integral, and simplify the resulting expression to a form that matches an integral formula from the table of integrals.
Use the appropriate integral formula from the table to evaluate the integral in terms of \(t\), then substitute back 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.
Integration Using Standard Integral Forms
Many integrals can be evaluated by recognizing their form and matching them to standard integral formulas found in tables. This approach simplifies complex integrals by avoiding lengthy derivations and directly applying known results.
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Circles in Standard Form Example 1
Substitution Method for Integrals
Substitution involves changing variables to simplify the integral into a standard form. By letting a new variable represent a complicated expression, the integral becomes easier to evaluate using known formulas or techniques.
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07:33Euler's Method
Handling Integrals Involving Square Roots and Rational Functions
Integrals with expressions like √(ax + b) and rational functions often require algebraic manipulation or substitution to simplify the root and denominator. Recognizing patterns and applying appropriate substitutions is key to solving these integrals.
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07:01Integrals Involving Natural Logs: Substitution
Related Practice
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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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
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
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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