5.5 The Substitution Rule/15

${\begin{aligned}\ \int sin(\pi t)dt\end{aligned}}$

${\begin{aligned}let\;u&=\pi t\\[2ex]du&=\pi dx\\[2ex]{\frac {1}{\pi }}du&=dx\end{aligned}}$

'${\begin{aligned}\ \int sin(\pi t)dt=\int sin(u)({\frac {1}{\pi }}du)=\int {\frac {1}{\pi }}(-cosu)+c=\int -{\frac {1}{\pi }}cos(\pi t)+c\end{aligned}}$