Example 3. Find $\int \cos ^{3} x \cdot \sin x d x$.
Solution. We notice that $\sin x d x$ is the differential of cosine (up to a sign)

$$
\int \cos ^{3} x \sin x d x=-\int \cos ^{3} x d(\cos x)=-\frac{1}{4} \cos ^{4} x+C
$$

Here, the economy is achieved by not writing out the intermediate integral $\int u^{3} d u$ and its result $\frac{u^{4}}{4}$.