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Chapter 6: Problem 5

Find all antiderivatives of each following function: $$f(x)=3$$

Short Answer

Expert verified
The antiderivatives of 3 are given by \( 3x + C \), where \( C \) is the constant of integration.

Step by step solution

01

Identify the given function

The given function is a constant function: \( f(x) = 3 \)
02

Recall the antiderivative of a constant

The antiderivative of a constant \(a\) is given by \( ax + C \), where \(C\) is the constant of integration.
03

Apply the antiderivative formula

For the given function \( f(x) = 3 \), the antiderivative is obtained by multiplying the constant with \(x\) and adding the constant of integration: \( \text{Antiderivative} = 3x + C \)
04

Write the general form of the antiderivative

The general form of the antiderivative for the function is: \( F(x) = 3x + C \)

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Key Concepts

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

Constant Function
A constant function is one that does not change, regardless of the input value. In simpler terms, no matter what value of x you plug in, you'll always get the same output. An example of a constant function is given in the exercise as 3. This means for any value of x, f(x) will always remain 3.

When dealing with constant functions in calculus, finding the antiderivative is relatively straightforward compared to variable-dependent functions.
Integration
Integration is the process of finding the antiderivative or the integral of a function. In simpler terms, it's like the opposite of differentiation. While differentiation involves finding the rate of change, integration focuses on accumulating the total change. This is especially useful in areas like finding area under a curve or solving differential equations.

In our exercise, we integrated the constant function f(x) = 3 and found its antiderivative.
Constant of Integration
The constant of integration, noted as C, is an essential part of finding antiderivatives. When performing integration, since differentiation erases constants (because the derivative of a constant is zero), there are infinitely many potential antiderivatives that differ by a constant.

Thus, after integrating, we add +C to our result, serving as a placeholder for any constant value. In our exercise, it's represented in the final antiderivative: 3x + C.
Antiderivative Formula
The antiderivative formula for a constant function takes the general form of ax + C. This formula is derived from the basic rules of integration. Here, 'a' represents the constant in the function, 'x' represents the variable, and 'C' is the constant of integration.

In the exercise, where our function f(x) = 3, the antiderivative formula is applied as follows: we multiply the constant 3 by x and then add the constant of integration. Thus, the result is 3x + C.

This approach provides a simple and effective way to handle constant functions.

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Most popular questions from this chapter

Find the value of \(k\) that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How?] $$\int 2 e^{4 x-1} d x=k e^{4 x-1}+C$$

Find the value of \(k\) that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How?] $$\int \frac{7}{(8-x)^{4}} d x=\frac{k}{(8-x)^{3}}+C$$

Find the area under each of the given curves. $$y=4 x ; x=2 \text { to } x=3$$

An investment grew at an exponential rate \(R(t)=700 e^{0.07 t}+1000,\) where \(t\) is in years and \(R(t)\) is in dollars per year. Approximate the net increase in value of the investment after the first 10 years (as \(t\) varies from 0 to 10 ).

Find the value of \(k\) that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How?] $$\int(3 x+2)^{4} d x=k(3 x+2)^{5}+C$$
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