5.5 The Substitution Rule/5

${\begin{aligned}\int \cos ^{3}{(\theta )}\sin {(\theta )}d{(\theta )}{\text{,}}\quad u=\cos {(\theta )}\\[2ex]\end{aligned}}$

${\begin{aligned}u&=\cos {(\theta )}\\[2ex]du&=-\sin {(\theta )}d{(\theta )}\\[2ex]-du&=\sin {(\theta )}d{(\theta )}\end{aligned}}$

${\begin{aligned}\int \cos ^{3}{(\theta )}\sin {(\theta )}d{(\theta )}&=-\int u^{3}du\\[2ex]&={\frac {-u^{4}}{4}}+C={\frac {-\cos ^{4}{(\theta )}}{4}}+C\\[2ex]&={\frac {-1}{4}}\cos ^{4}{(\theta )}+C\end{aligned}}$