Understanding Antiderivatives | Uniqueness and the Role of Arbitrary Constants

Antiderivatives are unique

In mathematics, an antiderivative, also known as the indefinite integral, is a function that can be differentiated to give the original function

In mathematics, an antiderivative, also known as the indefinite integral, is a function that can be differentiated to give the original function. It is the inverse operation of differentiation. More formally, if F(x) is an antiderivative of f(x), then the derivative of F(x) with respect to x is equal to f(x), denoted as:

d/dx [F(x)] = f(x)

Now, coming to the statement “Antiderivatives are unique,” this is not completely accurate. While antiderivatives are unique up to a constant, there can be multiple antiderivatives for a given function.

Let’s consider an example:

If f(x) = 2x, the antiderivative (indefinite integral) of f(x) with respect to x would be F(x) = x^2 + C, where C is a constant. Here, F(x) is an antiderivative of f(x), but there are actually infinitely many antiderivatives because we can add any constant value to the result.

So, in general, we can say that antiderivatives are unique up to an arbitrary constant, denoted as C. This constant arises because when we differentiate a constant, it becomes zero. Hence, adding or subtracting a constant to the antiderivative does not affect the derivative of the function.

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