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Chapter 16: Problem 24

Is either of the following equations correct? Give reasons for your answers. $$\text { a. } \frac{1}{\cos x} \int \cos x \, d x=\tan x+C$$ $$\text { b. } \frac{1}{\cos x} \int \cos x \, d x=\tan x+\frac{C}{\cos x}$$

Short Answer

Expert verified
Equation a is incorrect; equation b is correct.

Step by step solution

01

Understanding the Problem

We need to determine whether the given equations are correct by evaluating the integrals and comparing them to the right sides of the equations. Each equation describes a relationship involving the integral of a function and a constant.
02

Evaluate the Integral

Calculate the integral: \[ \int \cos x \, dx = \sin x + C \] This integral is found using basic integration rules where the integral of \( \cos x \) is \( \sin x \) plus a constant of integration \( C \).
03

Analyze Statement a

For equation a, transform the left-hand side using the evaluated integral: \[ \frac{1}{\cos x} \int \cos x \, dx = \frac{1}{\cos x} (\sin x + C) = \tan x + \frac{C}{\cos x} \] Therefore, it does not equal \( \tan x + C \) as provided in equation a.
04

Analyze Statement b

For equation b, the equivalent transformation shows: \[ \frac{1}{\cos x} \int \cos x \, dx = \tan x + \frac{C}{\cos x} \] This matches the given statement exactly, making equation b correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration
Integration is a fundamental concept in calculus that involves finding the anti-derivative or integral of a function. It is the reverse process of differentiation.
In integration, we seek to find a function whose derivative gives the original function. This can help us compute areas under curves, solve differential equations, and evaluate accumulated quantities.
For example, in the problem above, we evaluate the integral of \( \cos x \) using the formula:
  • The integral of \( \cos x \) is \( \sin x + C \), where \( C \) represents the constant of integration.
  • Integration involves recognizing patterns and using known integral formulas to solve problems efficiently.
Trigonometric Functions
Trigonometric functions are vital in calculus for describing relationships in geometry involving angles and sides of triangles, and they appear frequently in integration problems.
These functions, including sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), offer a means to express periodic phenomena such as waves and oscillations.
When integrating trigonometric functions:
  • Remember that the integral of \( \cos x \) is \( \sin x \), and the integral of \( \sin x \) is \(-\cos x \).
  • Use identities such as \( \tan x = \frac{\sin x}{\cos x} \) to transform expressions when necessary.
In the exercise provided, understanding the trigonometric function \( \cos x \) and its integration helps verify the equations presented. This knowledge is useful for precision in solving integrals involving these functions.
Integration Constants
Integration constants play a crucial role when finding indefinite integrals, as they account for any constant value that differentiates similarly.
When you integrate a function, you must always add the constant of integration \( C \) because the process of differentiation removes constant terms. Therefore, the antiderivative could differ by any constant.
In the context of the exercise:
  • Both provided equations feature \( C \), ensuring that they include the general solution of the integral since it could be shifted by some constant.
  • In statement b, \( \frac{C}{\cos x} \) is an expression involving the constant \( C \), reinforcing the need to consider constants carefully when manipulating equations.
This complexity highlights the importance of incorporating \( C \) in integrations so that the complete set of solutions is captured.

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Most popular questions from this chapter

Use a CAS to explore graphically each of the differential equations. Perform the following steps to help with your explorations. a. Plot a slope field for the differential equation in the given \(x y\) -window. b. Find the general solution of the differential equation using your CAS DE solver. c. Graph the solutions for the values of the arbitrary constant \(C=-2,-1,0,1,2\) superimposed on your slope field plot. d. Find and graph the solution that satisfies the specified initial condition over the interval \([0, b]\) e. Find the Euler numerical approximation to the solution of the initial value problem with 4 subintervals of the \(x\) -interval, and plot the Euler approximation superimposed on the graph produced in part (d). f. Repeat part (e) for \(8,16,\) and 32 subintervals. Plot these three Euler approximations superimposed on the graph from part (e). g. Find the error ( \(y\) (exact) \(-y\) (Euler)) at the specified point \(x=b\) for each of your four Euler approximations. Discuss the improvement in the percentage error. $$\begin{array}{l}y^{\prime}=(\sin x)(\sin y), \quad y(0)=2 ; \quad-6 \leq x \leq 6, \quad-6 \leq y \leq 6 \\\b=3 \pi / 2\end{array}$$

In 1925 Lotka and Volterra introduced the predator-prey equations, a system of equations that models the populations of two species, one of which preys on the other. Let \(x(t)\) represent the number of rabbits living in a region at time \(t,\) and \(y(t)\) the number of foxes in the same region. As time passes, the number of rabbits increases at a rate proportional to their population, and decreases at a rate proportional to the number of encounters between rabbits and foxes. The foxes, which compete for food, increase in number at a rate proportional to the number of encounters with rabbits but decrease at a rate proportional to the number of foxes. The number of encounters between rabbits and foxes is assumed to be proportional to the product of the two populations. These assumptions lead to the autonomous system $$\begin{aligned}\frac{d x}{d t} &=(a-b y) x \\\\\frac{d y}{d t} &=(-c+d x) y \end{aligned}$$ where \(a, b, c, d\) are positive constants. The values of these constants vary according to the specific situation being modeled. We can study the nature of the population changes without setting these constants to specific values. Show that (0,0) and \((c / d, a / b)\) are equilibrium points. Explain the meaning of each of these points.

Solve the differential equations $$e^{2 x} y^{\prime}+2 e^{2 x} y=2 x$$

Solve the differential equations $$x \frac{d y}{d x}=\frac{\cos x}{x}-2 y, \quad x>0$$

The autonomous differential equations represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0) .\) Which equilibria are stable, and which are unstable? $$\frac{d P}{d t}=2 P(P-3)$$
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