Chapter 7, Problem 1RE

To determine

Whether the given statement is true or not.

Answer to Problem 1RE

The given statement is true.

Explanation of Solution

Given:

“The indefinite integral is another term for the family of all anti-derivatives of a function”.

Definition used:

Anti-derivative:

“The reverse process of obtaining the derivative is called anti-differentiation.

That is, $F^{\prime }\left(x\right)=f\left(x\right)$, then $F\left(x\right)$ is an anti-derivative of $f\left(x\right)$”.

Description:

Suppose $F^{\prime }\left(x\right)=f\left(x\right)$.

By anti-derivative definition, $F\left(x\right)$ is an anti-derivative of $f\left(x\right)$.

Integrate the expression $F^{\prime }\left(x\right)=f\left(x\right)$ with respect to x,

$\begin{array}{l}{\displaystyle \int F^{\prime }\left(x\right)}={\displaystyle \int \left(f\left(x\right)\right)}dx\\ F\left(x\right)+C={\displaystyle \int f\left(x\right)}dx\end{array}$

That is, ${\displaystyle \int f\left(x\right)}dx=F\left(x\right)+C$.

This implies, ${\displaystyle \int f\left(x\right)}dx$ is the family of all the antiderivatives.

Thus, the family of all anti-derivatives of the function is the indefinite integral.