5.5 The Substitution Rule/5

\begin{align} Failed to parse (syntax error): {\displaystyle \int \cos^{3}{(\theta)}\sin{(\theta)}d{(\theta)} \text{,} \quad u=\cos{(\theta)}\\[2ex] } \end{align}

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

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