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4. What substitutions are made to evaluate integrals of sin(mx)sin(nx), sin(mx)cos(nx), and cos(mx)cos(nx)? Give an example of each case.
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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.PE.34
Evaluate the integrals in Exercises 33–36.
∫ [1 / (x(9 - x²))] dx
Guida verificata passo dopo passo1
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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Percorso guidato
10:07
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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6:04Intro 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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7:30Logarithms Introduction
Pratica correlata
Domanda del libro di testo
Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^(-x), and the line x = 1.
a. About the y-axis.
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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
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -2 to 0 of (x² - 1) dx
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