5.5 The Substitution Rule/5

$\int \cos ^{3}{(\theta )}\sin {(\theta )}d{(\theta )}{\text{,}}\quad u=\cos {(\theta )}$

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

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

&= \frac{-u^{4}}{4} + C = \frac{-\cos^{4}{(\theta)}}{4} + C &= \frac{-1}{4}\cos^{4}{(\theta)} + C