Chapter 15, Problem 1RE

Determine whether each of the following statements is true or false, and explain why.

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

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.