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Ch. 9 - First-Order Differential Equations
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 9 - First-Order Differential EquationsProblem 9.2.24
Chapter 9, Problem 9.2.24

Is either of the following equations correct? Give reasons for your answers.


a. (1/cosx) ∫ cos x dx = tan x + C
b. (1/cosx) ∫ cos x dx = tan x + C / cos x

Verified step by step guidance
1
Start by evaluating the integral \( \int \cos x \, dx \). Recall that the integral of \( \cos x \) is \( \sin x + C \), where \( C \) is the constant of integration.
Rewrite the left side of both equations by substituting the integral: \( \frac{1}{\cos x} \int \cos x \, dx = \frac{1}{\cos x} (\sin x + C) \). This simplifies to \( \frac{\sin x}{\cos x} + \frac{C}{\cos x} \).
Recognize that \( \frac{\sin x}{\cos x} = \tan x \), so the expression becomes \( \tan x + \frac{C}{\cos x} \).
Compare this result with the right sides of the given equations: (a) \( \tan x + C \) and (b) \( \tan x + \frac{C}{\cos x} \). Notice that the constant term in (a) does not match the derived expression, while (b) matches exactly.
Conclude that equation (b) is correct because it properly accounts for the constant of integration divided by \( \cos x \), whereas equation (a) incorrectly treats the constant as a simple additive constant without division by \( \cos x \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration of Basic Trigonometric Functions

Understanding how to integrate basic trigonometric functions like cos x is essential. The integral of cos x with respect to x is sin x plus a constant of integration, C. This fundamental fact helps evaluate the given expressions correctly.
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Introduction to Trigonometric Functions

Algebraic Manipulation of Expressions

After integrating, it is important to correctly manipulate algebraic expressions, especially when dividing by functions like cos x. This includes distributing division over sums and understanding how constants of integration behave under division.
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Simplifying Trig Expressions

Properties of the Constant of Integration

The constant of integration, C, represents an arbitrary constant and must be handled carefully. When dividing an integral expression by a function, the constant remains arbitrary and should not be combined or altered incorrectly, as it represents an infinite family of antiderivatives.
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Properties of Functions
Related Practice
Textbook Question

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.


y' = x(1-y), y(1) = 0, dx = 0.2

Textbook Question

Use Euler’s method with dx = 1/3 to estimate y(2) if y′ = x sin y and y(0) = 1. What is the exact value of y(2)?

Textbook Question

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.


dy/dx + xy = x, y(0) = -6

Textbook Question

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.


y' = y²(1+2x), (y-1) = 1, dx = 0.5

Textbook Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


e²ˣy' + 2e²ˣ y = 2x

Textbook Question

In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.


y' = 1 + y², y(0) = 0, dx = 0.1, x* = 1

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