Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² dx / √|x − 1|
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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 8.2.26
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ from 0 to 1 x√(1 - x) dx
Verified step by step guidance1
Identify the integral to solve: \(\int_0^1 x \sqrt{1 - x} \, dx\).
Use substitution to simplify the integral. Let \(u = 1 - x\), which implies \(du = -dx\) and \(x = 1 - u\).
Change the limits of integration according to the substitution: when \(x = 0\), \(u = 1\); when \(x = 1\), \(u = 0\).
Rewrite the integral in terms of \(u\): \(\int_1^0 (1 - u) \sqrt{u} (-du)\), then reverse the limits to get rid of the negative sign, resulting in \(\int_0^1 (1 - u) \sqrt{u} \, du\).
Expand the integrand to \(\int_0^1 (1 - u) u^{1/2} \, du = \int_0^1 (u^{1/2} - u^{3/2}) \, du\), and then integrate term-by-term.
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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. It involves choosing a substitution u = g(x) that simplifies the integrand, then rewriting the integral in terms of u and du. This technique is especially useful when the integral contains a composite function.
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Euler's Method
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is key to simplifying the integral effectively.
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06:18Integration by Parts for Definite Integrals
Definite Integrals and Limits of Integration
Definite integrals calculate the net area under a curve between two points, using specified limits of integration. When performing substitution, the limits must be adjusted to the new variable or the integral must be converted back to the original variable before evaluating. Proper handling of limits ensures accurate evaluation of the integral.
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05:43Definition of the Definite Integral
Related Practice
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.
∫₋₁¹ (√(1 + x²) sin x) dx
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Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ eˣ sec³(eˣ) dx
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.
∫ (6 dy / √y(1 + y))
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
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ tan^(-1)(x) / x² dx