Skip to main content
Contents Embed Dark Mode Prev Up Next \(\require{physics}\require{upgreek}\everymath{\displaystyle}
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\inv}{^{-1}}
\newcommand{\DS}{\displaystyle}
\newcommand{\eps}{\varepsilon}
\newcommand{\tsub}[1]{_{\mathrm{#1}}}
\newcommand{\ee}{\mathrm{e}}
\newcommand{\ii}{\mathrm{i}}
\newcommand{\limit}[1]{\lim\limits_{#1}}
\newcommand{\resid}[1]{\underset{#1}{\Res}}
\DeclareMathOperator{\sinc}{sinc}
\DeclareMathOperator{\sgn}{sgn}
\newcommand{\pii}{\pi}
\DeclareMathOperator{\Prob}{P}
\DeclareMathOperator{\EV}{E}
\DeclareMathOperator{\Var}{Var}
\newcommand{\bv}[1]{\boldsymbol{#1}}
\newcommand{\uv}[1]{\hat{\bv{#1}}}
\newcommand{\cl}[1]{\mathcal{#1}}
\newcommand{\bb}[1]{\mathbb{#1}}
\DeclareMathOperator{\Cis}{cis}
\DeclareMathOperator{\RE}{Re}
\DeclareMathOperator{\IM}{Im}
\newcommand{\xd}{\mathbf{d}}
\newcommand{\seq}[3]{{#1}_{#2},\cdots,{#1}_{#3}}
\newcommand{\psup}[1]{^{(#1)}}
\newcommand{\hypext}{{}^*}
\DeclareMathOperator{\st}{st}
\newcommand{\set}[1]{\left\{#1\right\}}
\DeclareMathOperator{\Sin}{Sin}
\DeclareMathOperator{\Cos}{Cos}
\DeclareMathOperator{\Tan}{Tan}
\DeclareMathOperator{\Sec}{Sec}
\DeclareMathOperator{\Csc}{Csc}
\DeclareMathOperator{\Cot}{Cot}
\DeclareMathOperator{\Log}{Log}
\DeclareMathOperator{\Arg}{Arg}
\DeclareMathOperator{\Ln}{Ln}
\DeclareMathOperator{\Grad}{grad}
\DeclareMathOperator{\Div}{div}
\DeclareMathOperator{\Curl}{curl}
\newcommand{\rd}{\textstyle\mathop{}\!\mathrm{d}^{\!\!\!-}\hspace{-0.0555 em}}
\newcommand{\rpd}{\textstyle\mathop{}\!\partial^{\hspace{-0.5 em}-}\hspace{-0.1666 em}}
\newcommand{\rdv}[2]{\frac{\rd{#1}}{\rd{#2}}}
\newcommand{\rpdv}[2]{\frac{\rpd{#1}}{\rpd{#2}}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 9.1 Functions defined by integrals
Objectives
Define new functions using integrals when no closed-form antiderivative exists.
Recover properties of the natural logarithm from its integral definition.
Examine examples of special functions from number theory and physics.
One way of defining the natural logarithm is
\begin{equation*}
\ln x=\int_{1}^{x}\frac{1}{t}\dd{t}\text{.}
\end{equation*}
You can then show that this function has all the properties you’d expect of the natural logarithm.
Many functions don’t have closed-form antiderivatives! But we do something very similar to deal with them—we define a new function. A few examples:
