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 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 8.1.26
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))
Verified step by step guidance1
Rewrite the integral to make it clearer: \(\int \frac{6}{\sqrt{y}(1 + y)} \, dy\).
Express \(\sqrt{y}\) as \(y^{1/2}\) and rewrite the integral as \(\int \frac{6}{y^{1/2}(1 + y)} \, dy\).
Consider the substitution \(y = t^2\), so that \(dy = 2t \, dt\) and \(\sqrt{y} = t\).
Rewrite the integral in terms of \(t\): replace \(y\) and \(dy\) accordingly, then simplify the expression.
After substitution, simplify the integral and look for a method to integrate, such as partial fractions or a direct algebraic simplification.
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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 involves changing variables to simplify an integral. By letting a new variable represent a part of the integrand, the integral can become easier to evaluate. This method is especially useful when the integrand contains composite functions or expressions under radicals.
Recommended video:
04:27
Substitution With an Extra Variable
Algebraic Manipulation of Integrands
Algebraic manipulation includes rewriting the integrand to a more convenient form, such as factoring, expanding, or simplifying expressions. This step can reveal standard integral forms or make substitution more straightforward, facilitating the integration process.
Recommended video:
05:22Completing the Square to Rewrite the Integrand
Properties of Radicals and Exponents
Understanding how to handle radicals and fractional exponents is crucial when integrating functions involving roots. Converting radicals to fractional powers allows the use of power rule integration, and recognizing how to simplify expressions under the root helps in choosing the right substitution.
Recommended video:
06:21Properties of Functions
Related Practice
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² dx / √|x − 1|
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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
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
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ tan^(-1)(x) / x² dx