Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int 6 d x$$

Step-by-Step Solution

Answer

The indefinite integral of the given function is: \(\int 6 dx = 6x + C\)

Step 1: Identifying the function to integrate

We are given the function $$\int 6 dx$$ which is just a constant function of 6.

Step 2: Finding the antiderivative

To find the indefinite integral of the constant function 6, we need to identify a function whose derivative with respect to x is 6. The derivative of a linear function of the form $$f(x) = Cx$$, where \(C\) is a constant, is just the constant \(C\). Therefore, if the derivative is equal to 6, we can deduce that the function must be of the form $$f(x) = 6x$$.

Step 3: Remember the constant of integration

When finding an indefinite integral, there is always a constant of integration, which we usually represent as C. This is because the derivative of a constant is zero, which means that any constant could be added to our function without changing its derivative. Thus, the most general antiderivative of our given function is: $$6x + C$$

Step 4: Write the final answer

The indefinite integral of our given function is: $$\int 6 dx = 6x + C$$

Key Concepts

Antiderivative

In calculus, an antiderivative is a function whose derivative is the given function. This fascinating concept arises whenever we want to reverse the process of differentiation. Finding the antiderivative is crucial when solving indefinite integrals.

• When given a function, like 6 in our example, we're essentially asked to find another function whose derivative is 6.
• The principle of finding an antiderivative relies on recognizing patterns from derivatives. For a constant function like 6, the antiderivative is the simple linear function 6x.

The understanding of this reverse process helps us to calculate areas and solve numerous problems in physics and engineering by returning us to the original function.

Constant of Integration

The constant of integration is often denoted as C, and it's nearly always part of the solution to an indefinite integral. This constant arises from the fact that the differentiation of any constant is zero.

• While the derivative of 6x is 6, we can add any constant value to 6x, such as 6x + 7 or 6x - 4, and the derivative will still be 6.
• This means there's an infinite number of functions that can serve as antiderivatives for 6, each differing by a constant value.

Hence, when finding antiderivatives, we add the constant of integration, C, to capture that infinite set of possibilities. Always remember, though, without this constant, the solution is incomplete!

Derivative of a Function

The derivative of a function gives us a measure of how a function changes as its input changes. It's one of the core concepts in calculus, providing us with the rate at which a function's value is changing at any given point.

• The derivative of a constant function, like 6, is essentially zero because it doesn't change with respect to its variable.
• For a linear function like 6x, the derivative with respect to x is simply 6, representing a constant rate of change.

Understanding derivatives allows us to track dynamic variations, which is why they are foundational for grasping more complex topics like the antiderivative and the constant of integration.