∫ cos 3 ( θ ) sin ( θ ) d ( θ ) , u = cos ( θ ) {\displaystyle {\begin{aligned}\int \cos ^{3}{(\theta )}\sin {(\theta )}d{(\theta )}{\text{,}}\quad u=\cos {(\theta )}\\[2ex]\end{aligned}}}
u = cos ( θ ) d u = − sin ( θ ) d ( θ ) − d u = sin ( θ ) d ( θ ) {\displaystyle {\begin{aligned}u&=\cos {(\theta )}\\[2ex]du&=-\sin {(\theta )}d{(\theta )}\\[2ex]-du&=\sin {(\theta )}d{(\theta )}\end{aligned}}}
∫ cos 3 ( θ ) sin ( θ ) d ( θ ) = − ∫ u 3 d u = − u 4 4 + C = − cos 4 ( θ ) 4 + C = − 1 4 cos 4 ( θ ) + C {\displaystyle {\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}}}
![{\displaystyle {\begin{aligned}\int \cos ^{3}{(\theta )}\sin {(\theta )}d{(\theta )}{\text{,}}\quad u=\cos {(\theta )}\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f198fb3b2b85e8390336ee4c12c74b1106dc6e3c)
![{\displaystyle {\begin{aligned}u&=\cos {(\theta )}\\[2ex]du&=-\sin {(\theta )}d{(\theta )}\\[2ex]-du&=\sin {(\theta )}d{(\theta )}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e43c3b8a3210b14665689ce0e483eaeaf5313e33)
![{\displaystyle {\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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ce8e663abfaa4484542357c4b2320ac8d33408c1)