This page presents practice questions on the concepts and properties of antiderivatives in calculus[cite: 1]. The questions are designed to help you build a solid understanding of how antiderivatives work and how they relate to differentiation. Each solution is hidden inside a collapsible dropdown so you can practice independently.
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.
Differentiating \( F(x) + c \) gives
\[
\frac{d}{dx}\bigl(F(x) + c\bigr) = F'(x) = f(x).
\]
True or False. If \( F(x) \) is an antiderivative of \( f(x) \), then \[ \frac{1}{c} F(cx) \] is an antiderivative of \( f(cx) \), where \( c \neq 0 \).
Answer: True.
Let \( u = cx \). Differentiating,
\[
\frac{d}{dx}\left( \frac{1}{c} F(cx) \right)
= \frac{1}{c} \cdot c \cdot F'(u)
= f(cx).
\]
True or False. An antiderivative of \( f \) plus an antiderivative of \( g \) is an antiderivative of \( f + g \).
Answer: True.
If \( F' = f \) and \( G' = g \), then
\[
\frac{d}{dx}(F + G) = F' + G' = f + g.
\]
True or False. An antiderivative of \( f \) divided by an antiderivative of \( g \) is an antiderivative of \( \dfrac{f}{g} \).
Answer: False.
Differentiating \( \dfrac{F}{G} \) gives
\[
\frac{d}{dx}\left(\frac{F}{G}\right)
= \frac{F'G - FG'}{G^2},
\]
which is not equal to \( \dfrac{f}{g} \) in general.