∫ cos 3 ( θ ) sin ( θ ) d ( θ ) , u = cos ( θ ) {\displaystyle \int \cos ^{3}{(\theta )}\sin {(\theta )}d{(\theta )}{\text{,}}\quad u=\cos {(\theta )}}
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 {\displaystyle {\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
![{\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]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e978e43bf40ec33b0b3b2de78802af02db8d65d1)