5.5 The Substitution Rule/9

$\int (3x-2)^{20}dx$

${\begin{aligned}u&=3x-2\\[2ex]du&=3dx\\[2ex]{\frac {1}{3}}du&=dx\\[2ex]\end{aligned}}$

${\begin{aligned}\int u^{20}\cdot {\frac {1}{3}}du\\[2ex]&={\frac {1}{3}}\int u^{20}du&={\frac {1}{3}}\cdot {\frac {1}{20+1}}u^{20+1}&={\frac {1}{3}}\cdot {\frac {1}{21}}u^{21}&={\frac {1}{63}}u^{21}+c&={\frac {1}{63}}(3x-2)^{21}+c\end{aligned}}$

$={\frac {1}{63}}(3x^{2}1-2097152)+c$