Question

Why do two different antiderivatives of a function differ by a constant?

Short answer

Answer: Two different antiderivatives of a function differ by a constant because when finding the antiderivative, we add a constant of integration (C) that does not affect the original derivative. As per the fundamental theorem of calculus, the antiderivative of a zero function is a constant function, and thus the difference between two antiderivatives G(x) and F(x) of the same function, G(x) - F(x), is equal to a constant C.

Step-by-step solution

Define antiderivatives

An antiderivative of a function f(x) is a function F(x) such that the derivative of F(x) is equal to f(x), i.e., F'(x) = f(x). It is also known as an indefinite integral.

Explain the concept of the constant of integration

When we find the antiderivative of a function, we usually add a constant, known as the constant of integration (C), because when we take the derivative of the function, the constant will become zero. In other words, the derivative of a constant is always zero, so adding a constant to the antiderivative does not affect its derivative.

Consider two antiderivatives of the same function

Let's say G(x) and F(x) are two different antiderivatives of the same function f(x). That means G'(x) = f(x) and F'(x) = f(x).

Find the difference between the two antiderivatives

Since G'(x) = f(x) and F'(x) = f(x), then G'(x) - F'(x) = 0. Integrate both sides of the equation with respect to x, and we get (G(x) - F(x))' = 0.

Show the difference is a constant

As per the fundamental theorem of calculus, the antiderivative of a zero function is a constant function. So, if (G(x) - F(x))' = 0, then G(x) - F(x) = C, where C is a constant. Therefore, two different antiderivatives of a function differ by a constant C.