Answer
$$\int \sin^52xdx=-\frac{\sin^42x\cos2x}{10}-\frac{2\sin^22x\cos2x}{15}-\frac{4}{15}\cos2x+C$$
Work Step by Step
$$A=\int \sin^52xdx$$
Use Reduction Formula 67, which states that
$$\int \sin^nax=-\frac{\sin^{n-1}ax\cos ax}{na}+\frac{n-1}{n}\int\sin^{n-2}axdx$$
for $a=2$ and $n=5$
$$A=-\frac{\sin^42x\cos2x}{10}+\frac{4}{5}\int\sin^32xdx$$
Use Reduction Formula 67 one more time, this time for $a=2$ and $n=3$
$$A=-\frac{\sin^42x\cos2x}{10}+\frac{4}{5}\Big(-\frac{\sin^22x\cos 2x}{6}+\frac{2}{3}\int\sin2xdx\Big)$$ $$A=-\frac{\sin^42x\cos2x}{10}+\frac{4}{5}\Big(-\frac{\sin^22x\cos 2x}{6}+\frac{2}{3}\times\Big(-\frac{1}{2}\Big)\cos2x\Big)+C$$ $$A=-\frac{\sin^42x\cos2x}{10}+\frac{4}{5}\Big(-\frac{\sin^22x\cos 2x}{6}-\frac{1}{3}\cos2x\Big)+C$$ $$A=-\frac{\sin^42x\cos2x}{10}-\frac{2\sin^22x\cos2x}{15}-\frac{4}{15}\cos2x+C$$
