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Chapter 5: Problem 25

Find the general antiderivative of the given function. $$ f(x)=\sin (2 x) $$

Short Answer

Expert verified
The general antiderivative of \( f(x) = \sin(2x) \) is \( F(x) = -\frac{1}{2}\cos(2x) + C \).

Step by step solution

01

Understand the Antiderivative Concept

An antiderivative of a function is a function whose derivative is the original function. For example, if \( F(x) \) is the antiderivative of \( f(x) \), then \( F'(x) = f(x) \).
02

Use the Integration Rule for Sine Function

We know from calculus that the integral of \( \sin(ax) \) is \( -\frac{1}{a}\cos(ax) + C \) where \( C \) is the constant of integration. This rule will help in finding the antiderivative of \( \sin(2x) \).
03

Apply the Integration Rule

Given \( f(x) = \sin(2x) \), apply the rule: \[\int \sin(2x) \, dx = -\frac{1}{2} \cos(2x) + C\]Here, \( a = 2 \), ensuring correct application of the antiderivative rule for sine functions.
04

Combine Constant of Integration

Every indefinite integral solution includes a constant \( C \) representing any constant value, hence the general antiderivative is:\[F(x) = -\frac{1}{2} \cos(2x) + C\]

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Key Concepts

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

Integration Rules
Integration rules are essential tools in calculus, enabling us to find antiderivatives of various functions. When we talk about integration rules, we're referring to a set of guidelines that simplify the process of integrating functions. In the context of this exercise, we are particularly interested in the integration rule for the sine function.

Here are some key points to keep in mind about integration rules:
  • Integration is the reverse process of differentiation. It involves finding the antiderivative or a function whose derivative is the original function.
  • Different functions have unique integration rules that must be applied correctly to obtain the accurate antiderivative.
  • These rules are derived from the properties of the functions and their derivatives, ensuring consistency across mathematical equations.
      • By understanding and applying integration rules correctly, we can easily determine antiderivatives and solve complex calculus problems efficiently. For the sine function, knowing that its antiderivative involves cosine is critical.
Sine Function
The sine function, denoted as \(\sin(x)\), is a fundamental trigonometric function that appears frequently in mathematical analyses involving periodic phenomena. It oscillates between -1 and 1 and has various important properties and uses in calculus.

When working with integrals involving the sine function, it's crucial to remember:
  • The sine function is periodic, with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) units.
  • Its derivative is the cosine function, \(\cos(x)\), while its antiderivative is \(-\cos(x)\) in the simplest case.
  • The integral of \(\sin(ax)\) leads to \(-\frac{1}{a} \cos(ax)\) when applying the integration rule for sine, as demonstrated in the provided solution.
In the given exercise, recognizing that the function is a sine function with a coefficient inside (like \(\sin(2x)\)) is essential. The understanding of its antiderivative is key to solving such problems accurately.
Constant of Integration
A fundamental aspect of indefinite integrals is the constant of integration, typically represented by \(C\). This constant acknowledges the fact that when taking derivatives, we lose track of constants since their derivative is zero.

Understanding the constant of integration includes:
  • Every indefinite integral solution can include multiple functions differing only by a constant, hence \(C\) accounts for these possible variations.
  • The value of \(C\) can be determined if additional conditions or information are provided, otherwise, it remains arbitrary in the general solution.
  • It ensures that the solution to the differential equation, represented by the indefinite integral, encompasses all possible antiderivatives.
In our antiderivative of \(f(x) = \sin(2x)\), we have \(F(x) = -\frac{1}{2} \cos(2x) + C\), showcasing the necessity of \(C\) in representing the most general form of the solution.

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Most popular questions from this chapter

Find the equilibria of $$x_{t+1}=\frac{x_{t}}{0.3+x_{t}}, \quad t=0,1,2, \ldots$$ and use the stability criterion for an equilibrium point to determine whether they are stable or unstable.

If a patient takes ibuprofen every \(T\) hours, rather than every 6 hours then the concentration of ibuprofen in their blood one hour after each pill is taken (that is, after \(1,1+T, 1+2 T\), hours, and so on) is given by a recurrence equation: $$C_{n+1}=(0.7575)^{T} C_{n}+40$$ (a) Find the equilibrium point of this recurence equation, and show that it is locally stable for any value of \(T>0\). (b) Assume that \(T=1\) and \(C_{1}=40 .\) Make a cobweb plot to illustrate the behavior of the sequence \(C_{1}, C_{2}, C_{3}, \ldots .\)

Find the general solution of the differential equation. $$ \frac{d y}{d x}=x(1+x), x>0 $$

(a) Use the stability criterion to characterize the stability of the equilibria of $$x_{t+1}=\frac{10 x_{t}^{2}}{9+x_{t}^{2}}, \quad t=0,1,2, \ldots$$ (b) Use cobwebbing to decide the limit \(x_{t}\) converges to as \(t \rightarrow \infty\) if (i) \(x_{0}=0.5\) and (ii) \(x_{0}=3\).

The Ricker model was introduced by Ricker (1954) as an alternative to the discrete logistic equation to describe the density-dependent growth of a population. Under the Ricker model the population \(N_{t}\) sampled at discrete times \(t=0,1,2, \ldots\) is modeled by a recurrence equation $$N_{t+1}=R_{0} N_{t} \exp \left(-a N_{t}\right)$$ where \(R_{0}\) and \(a\) are positive constants that will vary between different species and between different habitats. (a) Explain why you would expect \(R_{0}>1\) (Hint: consider the population growth when \(N_{t}\) is very small.) (b) Show that the recursion relation has two equilibria, a trivial equilibrium (that is, \(N=0\) ) and another equilibrium, which you should find. (c) Show that if \(R_{0}>1\) then use the stability criterion for equilibria to show that the trivial equilibrium point is unstable. (d) Use the stability criterion for equilibria to show that the nontrivial equilibrium point is stable if \(0<\ln R_{0}<2\). (e) If \(R_{0}>1\) then \(\ln R_{0}>0\), so most populations will meet the first inequality condition. What happens if \(\ln R_{0}>2 ?\) Let's try some explicit values: \(R_{0}=10, a=1, N_{0}=1 .\) Calculate the first ten terms of the sequence, and describe in words how the sequence behaves.
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