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Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 8, Problema 8.6.32
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ (√2 - x) / √x dx
Guida verificata passo dopo passo1
Identify a substitution that simplifies the integral. Since the integral contains both \( \sqrt{x} \) and \( x \), let’s set \( u = \sqrt{x} \). This means \( u = x^{1/2} \).
Express \( x \) and \( dx \) in terms of \( u \). Since \( u = x^{1/2} \), then \( x = u^2 \). Differentiate both sides with respect to \( u \) to find \( dx \): \( dx = 2u \, du \).
Rewrite the integral in terms of \( u \). Substitute \( \sqrt{x} = u \), \( x = u^2 \), and \( dx = 2u \, du \) into the integral: \[ \int \frac{\sqrt{2} - x}{\sqrt{x}} \, dx = \int \frac{\sqrt{2} - u^2}{u} \cdot 2u \, du \].
Simplify the integrand by canceling and combining terms: \( \frac{\sqrt{2} - u^2}{u} \cdot 2u = 2(\sqrt{2} - u^2) \). So the integral becomes \( \int 2(\sqrt{2} - u^2) \, du \).
Now, split the integral into two simpler integrals: \[ 2 \int \sqrt{2} \, du - 2 \int u^2 \, du \]. These are standard integrals that can be found in the integral table and evaluated easily.
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Substitution Method in Integration
The substitution method involves changing variables in an integral to simplify it into a more familiar form. By letting a new variable equal a function of the original variable, the integral can be rewritten and evaluated more easily, often matching standard integral forms.
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07:33
Euler's Method
Manipulating Radicals and Algebraic Expressions
Understanding how to rewrite expressions involving square roots and algebraic terms is essential. This includes expressing radicals as fractional exponents and simplifying the integrand to facilitate substitution and integration.
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06:13Limits of Rational Functions with Radicals
Using Integral Tables
Integral tables provide formulas for common integrals, allowing quick evaluation once the integral is transformed appropriately. Recognizing the form of the integral after substitution helps in matching it to a table entry for direct integration.
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Percorso guidato
08:01Integration Using Partial Fractions
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