Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x - x²) / x 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 8.6.56
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to 1/√3) dt / (t² + 1)^(7/2)
Verified step by step guidance1
Recognize that the integral has the form \(\int \frac{dt}{(t^2 + 1)^{7/2}}\), which suggests a trigonometric substitution because of the \(t^2 + 1\) term inside the power.
Use the substitution \(t = \tan(\theta)\), which implies \(dt = \sec^2(\theta) d\theta\) and \(t^2 + 1 = \sec^2(\theta)\).
Rewrite the integral in terms of \(\theta\): replace \(dt\) with \(\sec^2(\theta) d\theta\) and \((t^2 + 1)^{7/2}\) with \((\sec^2(\theta))^{7/2} = \sec^7(\theta)\), so the integrand becomes \(\frac{\sec^2(\theta)}{\sec^7(\theta)} = \sec^{-5}(\theta)\).
Change the limits of integration from \(t\) to \(\theta\): when \(t=0\), \(\theta = \arctan(0) = 0\); when \(t=\frac{1}{\sqrt{3}}\), \(\theta = \arctan\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}\).
Now the integral is \(\int_0^{\pi/6} \sec^{-5}(\theta) d\theta = \int_0^{\pi/6} \cos^5(\theta) d\theta\). Use a reduction formula for powers of cosine to evaluate this integral step-by-step.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving expressions like √(a² + x²), √(a² - x²), or √(x² - a²). By substituting x with a trigonometric function (e.g., x = a tan θ), the integral transforms into a trigonometric integral that is often easier to evaluate.
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Reduction Formulas
Reduction formulas express an integral with a certain power or parameter in terms of a similar integral with a lower power or simpler parameter. They help solve complex integrals step-by-step by breaking them down into simpler, more manageable integrals.
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Definite Integration with Limits
Definite integration involves evaluating the integral between specified limits. When using substitution, it is important to change the limits according to the substitution or revert to the original variable before applying the limits to find the exact numerical value.
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Related Practice
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x (7x + 5)^(3/2) dx
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ arcsin(y) dy
Textbook Question
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ csc³(√θ) / √θ dθ
Textbook Question
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ √(x² + 2x + 2) / (x² + 2x + 1) dx
Textbook Question
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = sin x, 0 ≤ x ≤ π
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