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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.34
Capitolo 8, Problema 8.PE.34

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

Guida verificata passo dopo passo
1
Start by recognizing that the integral involves a rational function with a quadratic expression in the denominator: \(\int \frac{1}{x(9 - x^{2})} \, dx\).
Use partial fraction decomposition to rewrite the integrand. Express \(\frac{1}{x(9 - x^{2})}\) as \(\frac{A}{x} + \frac{Bx + C}{9 - x^{2}}\).
Multiply both sides by the common denominator \(x(9 - x^{2})\) to get an equation without denominators: \(1 = A(9 - x^{2}) + (Bx + C)x\).
Expand and group like terms by powers of \(x\): \(1 = 9A - A x^{2} + B x^{2} + C x\).
Equate coefficients of corresponding powers of \(x\) on both sides to form a system of equations for \(A\), \(B\), and \(C\). Then solve this system to find the values of \(A\), \(B\), and \(C\).

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Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational functions into simpler fractions that are easier to integrate. It involves expressing the integrand as a sum of fractions with simpler denominators, often linear or quadratic, facilitating straightforward integration.
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Partial Fraction Decomposition: Distinct Linear Factors

Integration of Rational Functions

Integrating rational functions often requires algebraic manipulation such as partial fractions or substitution. Recognizing the form of the integrand helps determine the appropriate method, enabling the evaluation of integrals involving polynomials in numerator and denominator.
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Intro to Rational Functions

Logarithmic Integration

Logarithmic integration arises when integrating functions of the form 1/u, where u is a differentiable function of x. The integral results in a natural logarithm, ln|u| + C, which is common when integrating terms obtained after partial fraction decomposition.
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Logarithms Introduction
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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Evaluate the integrals in Exercises 33–36.

∫ [1 / √(9 - x²)] dx

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

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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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