Question

Describe the set of antiderivatives of \(f(x)=1\)

Short answer

Question: Describe the set of antiderivatives for the function f(x) = 1. Answer: The set of antiderivatives for the function f(x) = 1 is given by any linear function of the form x + C, where C is an arbitrary constant. This means that any vertical translation of the function x will be an antiderivative of f(x) = 1.

Step-by-step solution

Recognize the function and the need for an antiderivative

We are given the function f(x) = 1. Our goal is to find all possible antiderivatives of this function. The process of finding an antiderivative is called integration.

Integrate the function

To find the antiderivative of f(x) = 1, we use the indefinite integral notation: \(\displaystyle \int f(x) dx = \int 1 dx\)

Determine the antiderivative

Recall that the derivative of a constant function is always 0. However, in this case, the derivative of a linear function with a slope of 1 will give us the function f(x) = 1. The integration will yield a linear function in the form of \( x + C \), where \(C\) is an arbitrary constant. Thus, the antiderivative of f(x) = 1 is as follows: \(\displaystyle \int 1 dx = x + C\)

Describe the final result

The set of antiderivatives for the function f(x) = 1 is given by any linear function of the form x + C, where C is an arbitrary constant. In other words, any vertical translation of the function x will be an antiderivative of f(x) = 1.

Key concepts

Indefinite Integral

When we talk about antiderivatives, we're diving into the realm of indefinite integrals. An indefinite integral is essentially the reverse operation of differentiation. It is the process of finding a function whose derivative is the given function. The notation for an indefinite integral looks like this: \( \int f(x) \, dx \). This symbol represents the collection of all functions that can be differentiated to give the function \( f(x) \). Every result of an indefinite integral includes a constant, which we'll talk about more in another section. Indefinite integrals are crucial because they allow us to work backwards from a derivative to an antiderivative, gaining insight into the original function that led to the derivative.
When you integrate a simple function like \( f(x) = 1 \), it might seem straightforward. But remember, the idea is that every small piece of something you do to the function during integration can be reversed by differentiation.

Integration

Integration is the mathematical process used to find an antiderivative, or a function’s integral. Picture integration as a tool to "piece together" the original function from its derivative. There are two main kinds of integration: definite and indefinite. Here, we're interested in indefinite integration, which doesn't involve boundaries or limits.Even when the function is simple, like \( f(x) = 1 \), integration serves a big purpose. It tells us not just about the slope of a line, but about every possible way this slope could exist in a function. For \( f(x) = 1 \), you might find the integral \( \int 1 \, dx = x + C \). That tells us the set of functions whose derivative is \( 1 \) is any function in the form of \( x + C \).During the process of integration, you will come across different techniques depending on the complexity of the function. For simple functions like polynomials, integrating is relatively straightforward. But remember, the general approach stays consistent: you find a potential antiderivative and add a constant term to cover all potential solutions.

Constant of Integration

Every indefinite integral comes with a special little addition: the constant of integration, often represented by \( C \). But why do we need it? When integrating, you might think there's just one simple answer, but many functions can have the same derivative. This means for every derivative, there's actually a whole family of potential antiderivatives. Say you've integrated \( f(x) = 1 \) to get \( F(x) = x + C \). That \( C \) represents any constant number. It shows that adding any value to the function \( x \) gives us another valid function with the same derivative, which is why we call it the "constant of integration."In practice, the constant of integration is a reminder that without an initial condition or boundary details, you cannot pinpoint a problem's "exact" antiderivative. Instead, you get a whole set of them, all sharing the same basic shape but varying by a vertical shift. This small touch in your integral keeps your solution as general as possible.