Skip to main content\(\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 8.1 Integration by substitution
Objectives
-
Reverse the Chain Rule by introducing a substitution to simplify the integrand.
-
Adjust limits of integration after changing variables in definite integrals.
-
Use multi-stage substitutions for more complicated expressions.
-
This is the “Anti-Chain Rule”: making the substitution \(u=g(x)\) and \(\dd u=g'(x)\dd{x}\) gives
\begin{equation*}
\int f(g(x))g'(x)\dd{x}=\int f(u)\dd{u}
\end{equation*}
-
Transforming definite integrals:
Aside
\begin{equation*}
\int_{a}^{b} f(g(x))g'(x)\dd{x}=\int_{g(a)}^{g(b)}f(u)\dd{u}
\end{equation*}
-
Look at examples where multiple substitutions are needed, such as
\begin{equation*}
\int x\sqrt{x+1}\dd{x}
\end{equation*}
