To integrate the product of three cosine functions, we use the product-to-sum identity $2\cos A \cos B = \cos(A+B) + \cos(A-B)$, which is the cosine counterpart to the sine identity provided in the context. We simplify the expression by combining terms two at a time. First, we combine $\cos 4x$ and $\cos 6x$, then multiply by $\cos 2x$ and expand. The resulting terms are either linear cosines (which integrate to sines) or a $\cos^2$ term, which is integrated using the identity $\cos^2 \theta = \frac{1+\cos 2\theta}{2}$ found in the context.