Textbook Question
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ from 0 to 1 x√(1 - x) 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 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 8.6.20
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ tan^(-1)(x) / x² dx
Verified step by step guidance1
Recognize that the integral is \( \int \frac{\tan^{-1}(x)}{x^2} \, dx \), where \( \tan^{-1}(x) \) is the inverse tangent function, also written as \( \arctan(x) \).
Consider using integration by parts, since the integrand is a product of functions: one involving \( \arctan(x) \) and the other involving \( \frac{1}{x^2} \). Set \( u = \arctan(x) \) and \( dv = \frac{1}{x^2} dx \).
Compute \( du \) and \( v \):
- \( du = \frac{1}{1+x^2} dx \) because the derivative of \( \arctan(x) \) is \( \frac{1}{1+x^2} \).
- \( v = \int x^{-2} dx = -x^{-1} = -\frac{1}{x} \).
Apply the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
Substitute the expressions for \( u, v, du \) to rewrite the integral.
Simplify the resulting integral and evaluate it using the table of integrals if necessary, focusing on the integral \( \int \frac{1}{x(1+x^2)} dx \) that appears after substitution.
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7mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Trigonometric Functions
Inverse trigonometric functions, like arctan(x), are the inverses of the standard trig functions and return angles given a ratio. Understanding their properties and derivatives is essential for integrating expressions involving these functions.
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Derivatives of Other Inverse Trigonometric Functions
Integration Techniques
Integration techniques such as integration by parts or substitution are often required to solve integrals involving products or compositions of functions, especially when standard formulas are not directly applicable.
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06:18Integration by Parts for Definite Integrals
Use of Integral Tables
Integral tables provide formulas for common integrals, including those involving inverse trig functions. Knowing how to locate and apply these formulas can simplify the evaluation of complex integrals.
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Related Practice
Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ x² √(1 - x²) dx
Textbook Question
Average value
In a mass-spring-dashpot system like the one in Exercise 65, the mass's position at time t is
y = 4e^(-t)(sin(t) - cos(t)), t ≥ 0.
Find the average value of y over the interval 0 ≤ t ≤ 2π.
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ e^(z + eᶻ) dz
Textbook Question
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Use numerical integration to estimate the value of
arcsin(0.6) = ∫ (from 0 to 0.6) dx / √(1 - x²).
For reference, arcsin(0.6) = 0.64350 to five decimal places.
Textbook Question
Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.
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