Skip to main content
Contents Index Dark Mode Prev Up Next \(\newcommand{\dollar}{\$}
\DeclareMathOperator{\erf}{erf}
\DeclareMathOperator{\arctanh}{arctanh}
\DeclareMathOperator{\arcsec}{arcsec}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Activity 5.4.2 .
Evaluate each of the following indefinite integrals. Check each antiderivative that you find by differentiating.
(a)
\(\displaystyle \int te^{-t} \, dt\)
(b)
\(\displaystyle \int 4x \sin(3x) \, dx\)
(c)
\(\displaystyle \int z \sec^2(z) \,dz\)
Note : after applying integration by parts, we need to evaluate
\(\int \tan(z) \, dz\text{.}\) To do so, recall that
\(\tan(z) = \frac{\sin(z)}{\cos(z)}\) and think about a
\(u\) -substitution that leads to an antiderivative for
\(\tan(z)\text{.}\)
(d)
\(\displaystyle \int x \ln(x) \, dx\)
