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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.PE.36
Capitolo 8, Problema 8.PE.36

Evaluate the integrals in Exercises 33–36.
∫ [1 / √(9 - x²)] dx

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1
Recognize that the integral \( \int \frac{1}{\sqrt{9 - x^2}} \, dx \) resembles the standard form \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx \), where \( a = 3 \).
Recall the formula for this type of integral: \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C \), where \( C \) is the constant of integration.
Apply the formula by substituting \( a = 3 \) into the expression, giving \( \arcsin\left(\frac{x}{3}\right) + C \).
Write the final integral expression as \( \int \frac{1}{\sqrt{9 - x^2}} \, dx = \arcsin\left(\frac{x}{3}\right) + C \).
Remember to include the constant of integration \( C \) since this is an indefinite integral.

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Integration of Functions Involving Square Roots

Integrals containing expressions like √(a² - x²) often require recognizing standard forms or using trigonometric substitution. These integrals relate to inverse trigonometric functions, which simplify the integration process.
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Integrals Involving Natural Logs: Substitution

Inverse Trigonometric Functions

Inverse trigonometric functions such as arcsin, arccos, and arctan arise naturally when integrating functions involving square roots of quadratic expressions. For example, ∫ dx/√(a² - x²) equals arcsin(x/a) + C.
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Derivatives of Other Inverse Trigonometric Functions

Substitution Method in Integration

Substitution is a technique to simplify integrals by changing variables. For integrals like ∫ dx/√(9 - x²), substituting x = 3 sin θ transforms the integral into a simpler trigonometric form, making it easier to evaluate.
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Euler's Method
Pratica correlata
Domanda del libro di testo

Heat capacity of a gas

Heat capacity 

C_v

is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal/deg-mol (calories per degree gram molecular weight).

The heat capacity of oxygen depends on its temperature T and satisfies the formula

C_v = 8.27 + 10^(-5) * (26T − 1.87T²)

Use Simpson’s Rule to find the average value of C_v and the temperature at which it is attained for

20°C ≤ T ≤ 675°C.

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Domanda del libro di testo

Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.

∫ [x / √(4 − x²)] dx

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Domanda del libro di testo

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ sinx·cos²x dx

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Domanda del libro di testo

135. Evaluate ∫₀^(π/2) (sin x) / (sin x + cos x) dx in two ways:

(a) By evaluating ∫ (sin x) / (sin x + cos x) dx, then using the Evaluation Theorem.

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Domanda del libro di testo

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ t dt / √(9 − 4t²)

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Domanda del libro di testo

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [x / (x² + 4x + 3)] dx

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