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Chapter 4: Problem 4

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

Short Answer

Expert verified
Answer: Two antiderivatives of the same function, F(x) and G(x), differ by a constant because when we subtract one from the other, i.e., H(x) = G(x) - F(x), the derivative H'(x) = G'(x) - F'(x) becomes equal to zero. This implies that H(x) is a constant function, and thus G(x) = F(x) + C, where C is a constant.

Step by step solution

01

Definition of an Antiderivative

An antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x) for all values x in the domain of f(x). In other words, the derivative of F(x) is equal to f(x).
02

Relation Among Different Antiderivatives of a Function

Let's assume that G(x) and F(x) are two different antiderivatives of the same function f(x). Therefore, G'(x) = f(x) and F'(x) = f(x). We want to show that these two antiderivatives differ by a constant, i.e., we want to show that G(x) = F(x) + C for some constant C.
03

Finding the Difference Between the Two Antiderivatives

Let's define a new function H(x) that represents the difference between G(x) and F(x), i.e. H(x) = G(x) - F(x). If we can prove that H'(x) = 0, then we can conclude that H(x) is a constant function and thus G(x) and F(x) differ by a constant C.
04

Derivative of Function H(x)

Taking the derivative of H(x), we get: H'(x) = G'(x) - F'(x). As we know that G'(x) = f(x) and F'(x) = f(x), we have: H'(x) = f(x) - f(x). This simplifies to: H'(x) = 0.
05

Conclusion

Since we found H'(x) = 0, and H(x) = G(x) - F(x), we can conclude that H(x) is a constant function. That means G(x) - F(x) = C, where C is a constant. So, we can say that two different antiderivatives of a function differ by a constant.

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