Difference between integral function and antiderivative.

Question

The definition of antiderivative given in my book is:

Definition:A differentiable function $F$(if it exists) such that $F'=f$,then $F$ is called antiderivative of $f$

First part of fundamental theorem precisely says that if a function is continuous and defined on $[a,b]$ then integral function $\int_0^{x}f(x)dx$ is differentiable and is antiderivative of $f$.The second part of fundamental theorem of calculus says that:

Theorem: If $f:[a,b]\rightarrow\mathbb{R}$ be bounded and Riemann integrable function and $F$ be its antiderivative then $$\int_a^{b}f(x)dx=F(b)-F(a)$$

My question is whether the antiderivative $F$ given in above theorem is necessarily integral function(+some constant) i.e $F(x)=\int_a^{x}f(x)dx+C$?,where $C$ is some constant.If it is not so then give a example of bounded and Riemann integrable function whose antiderivative exists but is not equal to integral function $\int_a^{x}f(x)dx$.

Answer 1

Denote $G(x) = \int_a^{x}f(x)dx$

Obviously $G(x) = F(x) - F(a)$

(this follows from the theorem)

If you now take derivatives of $F$ and $G$ you will see they are the same.

So $F$ and $G$ differ by a constant.

So the answer to your question is positive.

Answer 2

A function $F(x)$ is called an antiderivative of a function of $f(x)$ if $F′( x) = f(x)$ for all $x$ in the domain of f. Note that the function $F$ is not unique and that an infinite number of antiderivatives could exist for a given function. For example, $F(x) = x^3$, $G( x) = x^3 + 5$, and $H(x) = x^3 − 2$ are all antiderivatives of $f(x) = 3x^2$ because $F′(x) = G′(x) = H′(x) = f(x)$ for all $x$ in the domain of $f$. It is clear that these functions $F$, $G$, and $H$ differ only by some constant value and that the derivative of that constant value is always zero. In other words, if $F(x)$ and $G(x)$ are antiderivatives of $f(x)$ on some interval, then $F′(x) = G′(x)$ and $F(x) = G(x) + C$ for some constant $C$ in the interval. Geometrically, this means that the graphs of $F(x)$ and $G(x)$ are identical except for their vertical position.

The notation used to represent all antiderivatives of a function $f(x)$ is the indefinite integral symbol written $\int$,$\int f(x) dx=F(x)+C$ , where .The function of $f(x)$ is called the integrand, and $C$ is refered to as the constant of integration. The expression $F(x) + C$ is called the indefinite integral of F with respect to the independent variable $x$. Using the previous example of $F(x) = x^3$ and f( x) = 3x^2, you find that when we take an indefinite integral, we are in reality finding “all” the possible antiderivatives at once (as different values of $C$ gives different antiderivatives)

The indefinite integral of a function is sometimes called the general antiderivative of the function as well. In additionally, we would say that a definite integral is a number which we could apply the second part of the Fundamental Theorem of Calculus; but an antiderivative is a function which we could apply the first part of the Fundamental Theorem of Calculus.