Textbook Question
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.
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.PE.34
Evaluate the integrals in Exercises 33–36.
∫ [1 / (x(9 - x²))] dx
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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.
Recommended video:
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.
Recommended video:
7:30Logarithms Introduction
Related Practice
Textbook Question
Using different substitutions
Show that the integral
∫((x² - 1)(x + 1))^(-2/3) dx
can be evaluated with any of the following substitutions.
a. u = 1/(x + 1)
What is the value of the integral?
Textbook Question
You are planning to use Simpson’s Rule to estimate the value of the integral Estimate ∫ from 1 to 2 of f(x) dx with an error magnitude less than 10⁻⁵ using Simpson’s Rule.
You have determined that |f⁽⁴⁾(x)| ≤ 3 throughout the interval of integration. How many subintervals should you use to ensure the required accuracy?
(Remember that for Simpson’s Rule the number of subintervals must be even.)
Textbook Question
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.
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (2 − cosx + sinx) / sin²x dx
Textbook Question
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 -1 to 1 of (x² + 1) dx