## Calculus, 10th Edition (Anton)

Changing $\sin^{2}(x)$ to $1-\cos^{2}(x)$ changes the integral to $$\int{\sin(x)(1-\cos^{2})^{2}\cos^{8}(x)}dx$$. Performing a u-substitution with $u=\cos(x)$ and $-du=\sin(x)dx$ leaves $$-\int{u^{8}(1-u^{2})^{2}}du$$ This can be expanded and solved using power rule.