Chapter 4 - Integration - 4.3 Integration By Substitution - Exercises Set 4.3 - Page 286: 25

$-\frac{\cos^5 3t}{15}+C$

Let $u=\cos 3t$, so $du=-3\sin 3tdt$. Then $\sin3tdt=-\frac{1}{3}du$. $\int\cos^4 3t\sin 3tdt$ $=\int u^4*(-\frac{1}{3})du$ $=-\frac{1}{3}*\frac{u^5}{5}+C$ $=-\frac{1}{3}*\frac{\cos^5 3t}{5}+C$ $=-\frac{\cos^5 3t}{15}+C$