# Antiderivatives in Calculus

Questions on the concepts and properties of antiderivatives in calculus are presented. These questions have been designed to help you **better understand the concept and properties of antiderivatives**. In order to answer the questions below, you first need to review the definitions and theorems related to antiderivatives.

## Questions with Solutions

### Question 1

**True or False**. If \( F(x) \) is an antiderivative of \( f(x) \) and \( c \) is any constant, then \( F(x) + c \) is also an antiderivative of \( f(x) \).

__Answer :__True.

Differentiate \( F(x) + c \).

### Question 2

**True or False**. If \( F(x) \) is an antiderivative of \( f(x) \), then

\( \dfrac{1}{c} F(cx) \) is an antiderivative of \( f(cx) \), where \( c \) is any non-zero constant.

__Answer :__True.

Let \( u = cx \) and differentiate \( \dfrac{1}{c} F(cx) \) with respect to \( x \):

\( \dfrac{d}{dx}\left(\dfrac{1}{c} F(cx)\right) \)

\( = \dfrac{1}{c} \dfrac{d(u)}{dx} \dfrac{dF}{du} \) (chain rule)

\( = \dfrac{1}{c} cf(u) = f(cx) \)

### Question 3

**True or False**. An antiderivative of function \( f \)

**plus**an antiderivative of function \( g \) is an antiderivative of function \( f + g \).

__Answer :__True.

Use the rule of differentiation to differentiate \( F + G \), where \( F \) is the antiderivative of \( f \) and \( G \) is the antiderivative of \( g \), and see that you can get \( f + g \).

### Question 4

**True or False**. An antiderivative of function \( f \)

**divided**by an antiderivative of function \( g \) is an antiderivative of function \( \dfrac{f}{g} \).

__Answer :__False.

Use the rule of differentiation to differentiate \( \dfrac{F}{G} \), where \( F \) is the antiderivative of \( f \) and \( G \) is the antiderivative of \( g \), and see that you cannot get \( \dfrac{f}{g} \).