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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.6.10
Capitolo 8, Problema 8.6.10

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x - x²) / x dx

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1
Start by rewriting the integrand to a more manageable form. Notice that the expression under the square root is \(x - x^2\), which can be factored as \(x(1 - x)\). So the integral becomes \(\int \frac{\sqrt{x(1 - x)}}{x} \, dx\).
Simplify the integrand by separating the square root and dividing by \(x\): \(\frac{\sqrt{x(1 - x)}}{x} = \frac{\sqrt{x} \sqrt{1 - x}}{x} = \frac{\sqrt{1 - x}}{\sqrt{x}}\).
Rewrite the integral as \(\int \frac{\sqrt{1 - x}}{\sqrt{x}} \, dx = \int \frac{(1 - x)^{1/2}}{x^{1/2}} \, dx = \int x^{-1/2} (1 - x)^{1/2} \, dx\).
To evaluate this integral, consider a substitution that simplifies the expression. A common substitution for integrals involving \(\sqrt{1 - x}\) is \(x = \sin^2 \theta\), because \(1 - \sin^2 \theta = \cos^2 \theta\). This substitution will transform the integral into a trigonometric integral.
After substituting \(x = \sin^2 \theta\), express \(dx\) in terms of \(d\theta\), rewrite the integral entirely in terms of \(\theta\), and then use the table of integrals to evaluate the resulting trigonometric integral.

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Integration Using Substitution

Substitution is a method to simplify integrals by changing variables, making the integral easier to evaluate. It often involves identifying a part of the integrand whose derivative also appears, allowing a direct replacement. This technique is essential when dealing with composite functions or expressions under roots.
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Substitution With an Extra Variable

Using Integral Tables

Integral tables provide formulas for common integrals, saving time and effort in manual integration. Recognizing the form of the integrand and matching it to a formula in the table is crucial. This skill helps in quickly evaluating integrals that might be complicated to solve from first principles.
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Integration Using Partial Fractions

Simplifying the Integrand

Before integrating, simplifying the integrand by algebraic manipulation or rewriting expressions can make the integral more manageable. For example, rewriting √(x - x²) as √(x(1 - x)) can suggest substitutions or reveal standard integral forms. Simplification is a key step in preparing the integral for substitution or table lookup.
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Completing the Square to Rewrite the Integrand
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