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 8.5.74a
Evaluate ∫ sec θ dθ by:
a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.
Verified step by step guidance1
Start with the integral: \(\int \sec \theta \, d\theta\).
Multiply the integrand by \(\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}\), which is equivalent to multiplying by 1, so the integral becomes \(\int \sec \theta \cdot \frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta} \, d\theta\).
Rewrite the numerator as \(\sec \theta (\sec \theta + \tan \theta) = \sec^2 \theta + \sec \theta \tan \theta\), so the integral is now \(\int \frac{\sec^2 \theta + \sec \theta \tan \theta}{\sec \theta + \tan \theta} \, d\theta\).
Let \(u = \sec \theta + \tan \theta\). Then compute \(\frac{du}{d\theta}\) by differentiating \(u\) with respect to \(\theta\): \(\frac{du}{d\theta} = \sec \theta \tan \theta + \sec^2 \theta\).
Notice that the numerator of the integrand matches \(\frac{du}{d\theta}\), so rewrite the integral as \(\int \frac{du}{u}\), which can be integrated using the natural logarithm function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Trigonometric Functions
This involves techniques to find antiderivatives of trigonometric expressions. Understanding how to manipulate functions like secant and tangent is essential for simplifying integrals and applying substitution methods effectively.
Recommended video:
6:04
Introduction to Trigonometric Functions
Multiplying by a Conjugate Expression
Multiplying the integrand by (sec θ + tan θ) / (sec θ + tan θ) is a strategic algebraic step that simplifies the integral. This technique leverages identities to rewrite the integral in a form that is easier to integrate.
Recommended video:
7:24Multiplying & Dividing Functions
U-Substitution Method
U-substitution is a method for integrating composite functions by substituting a part of the integrand with a new variable u. It simplifies the integral by transforming it into a basic form, often turning complicated trigonometric integrals into standard ones.
Recommended video:
04:27Substitution With an Extra Variable
Related Practice
Textbook Question
81. Find the values of p for which each integral converges.
a. ∫ from 1 to 2 of [dx / (x (ln x)^p)]
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
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 2 of 1/s² ds
Textbook Question
Finding area
Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:
a. π/2 ≤ x ≤ 3π/2.
1
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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