5.5 The Substitution Rule/5

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

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int \cos^{3}{(\theta)}\sin{(\theta)}d{(\theta)} &= \-int u^{3}du \\[2ex] &= \frac{-u^{4}}{4} + C = \frac{-\cos^{4}{(\theta)}}{4} + C \end{align} }

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