The relationship between definite integrals and antiderivatives
suggests a more convenient notation for the latter.
The indefinite integral of a function \(f,\)
denoted \(\int f(x)\,\mathrm{d}x\) (without bounds),
denotes the family of all antiderivatives of \(f.\)
The definite integral of a function on the interval \([a,b]\)
can be computed as the indefinite integral evaluated at the bounds \(x=a\) and \(x=b\)
Symbolically, for \(f(x) = F'(x),\)
\[
\int f(x)\,\mathrm{d}x \;=\; F(x) + C
\quad\qquad\implies\qquad\quad
\int_a^b f(x)\,\mathrm{d}x
%\;=\; \biggl(\int f(x)\,\mathrm{d}x\biggr)\biggr\rvert_{x=a}^{x=b}
\;=\; \Bigl(F(x)\!+\!C\Bigr)\biggr\rvert_{x=a}^{x=b}
\;=\; \Bigl(F(b)\!+\!C\Bigr)-\Bigl(F(a)\!+\!C\Bigr)
\;=\; F(b)\!-\!F(a)
\,.
\]
This indefinite integral notation affords us a more “forward” way
of writing antidifferentiation formulas:
\(\displaystyle \int x^{\color{Maroon}n} \,\mathrm{d}x = {\color{Maroon}\tfrac{1}{n+1}}x^{\color{Maroon}n+1} + C \)
The power rule for \(n \neq -1\)
\(\displaystyle \int \tfrac{1}{x} \,\mathrm{d}x = \ln(x) + C \)
The power rule for \(n = -1\)
\[\begin{align*}
\int {\color{DarkSlateBlue}\cos}(x) \,\mathrm{d}x
&= {\color{DarkGoldenRod}\sin}(x) + C
\\[1em]
\int {\color{DarkGoldenRod}\sin}(x) \,\mathrm{d}x
&= -{\color{DarkSlateBlue}\cos}(x) + C
\end{align*}\]
\[\begin{align*}
\int {\color{ForestGreen}\sec}(x){\color{Firebrick}\tan}(x) \,\mathrm{d}x
&= {\color{ForestGreen}\sec}(x) + C
\\[1em]
\int {\color{DarkOliveGreen}\csc}(x){\color{Maroon}\cot}(x) \,\mathrm{d}x
&= -{\color{DarkOliveGreen}\csc}(x) + C
\end{align*}\]
\[\begin{align*}
\int {\color{ForestGreen}\sec}^2(x) \,\mathrm{d}x
&= {\color{Firebrick}\tan}(x) + C
\\[1em]
\int {\color{DarkOliveGreen}\csc}^2(x) \,\mathrm{d}x
&= -{\color{Maroon}\cot}(x) + C
\end{align*}\]