Consider a function \(F\) and its derivative \(f\) — i.e. \(f = F'.\)
Since \(f\) is the derivative of \(F,\)
we refer to \(F\) as an antiderivative of \(f.\)
Since the operation of “taking the derivative” is referred to as differentiation,
we refer to “taking an antiderivative” as antidifferentiation.
A function’s derivative is unique, but it’s antiderivative is not.
A function has infinitely many antiderivatives,
all of them differing by some additive constant \(C.\)
Notationally, \(F\!+\!C\) is the family of all antiderivatives of \(f.\)
Whereas calculating the formula for a derivative is rote and procedural,
calculating a formula for an antiderivative may require some ingenuity,
and is sometimes not possible.
\(\displaystyle x^{\color{Maroon}n} = \frac{\mathrm{d}}{\mathrm{d}x}\Bigl( {\color{Maroon}\tfrac{1}{n+1}}x^{\color{Maroon}n+1} + C\Bigr) \)
The power rule for \(n \neq -1\)
\(\displaystyle \tfrac{1}{x} = \frac{\mathrm{d}}{\mathrm{d}x}\Bigl( \ln(x) + C\Bigr) \)
The power rule for \(n = -1\)
\[\begin{align*}
{\color{DarkSlateBlue}\cos}(x)
&= \frac{\mathrm{d}}{\mathrm{d}x}\Bigl( {\color{DarkGoldenRod}\sin}(x) + C \Bigr)
\\[1em]
{\color{DarkGoldenRod}\sin}(x)
&= \frac{\mathrm{d}}{\mathrm{d}x}\Bigl( -{\color{DarkSlateBlue}\cos}(x) + C \Bigr)
\end{align*}\]
\[\begin{align*}
{\color{ForestGreen}\sec}(x){\color{Firebrick}\tan}(x)
&= \frac{\mathrm{d}}{\mathrm{d}x}\Bigl( {\color{ForestGreen}\sec}(x) + C \Bigr)
\\[1em]
{\color{DarkOliveGreen}\csc}(x){\color{Maroon}\cot}(x)
&= \frac{\mathrm{d}}{\mathrm{d}x}\Bigl( -{\color{DarkOliveGreen}\csc}(x) + C \Bigr)
\end{align*}\]
\[\begin{align*}
{\color{ForestGreen}\sec}^2(x)
&= \frac{\mathrm{d}}{\mathrm{d}x}\Bigl( {\color{Firebrick}\tan}(x) + C \Bigr)
\\[1em]
{\color{DarkOliveGreen}\csc}^2(x)
&= \frac{\mathrm{d}}{\mathrm{d}x}\Bigl( -{\color{Maroon}\cot}(x) + C \Bigr)
\end{align*}\]