1. **Stating the problem:** Evaluate the integral $$\int 8 \cot(x) \cos(x)^2 \ dx$$. 2. **Recall definitions and formulas:** - The cotangent function is $$\cot(x) = \frac{\cos(x)}{\sin(x)}$$. - The integral involves trigonometric functions and powers. 3. **Rewrite the integral using the definition of cotangent:** $$\int 8 \cot(x) \cos(x)^2 \ dx = \int 8 \frac{\cos(x)}{\sin(x)} \cos(x)^2 \ dx = \int 8 \frac{\cos^3(x)}{\sin(x)} \ dx$$ 4. **Substitution:** Let $$u = \sin(x)$$, then $$du = \cos(x) dx$$. 5. **Rewrite the integral in terms of $$u$$:** We have $$\cos^3(x) dx = \cos^2(x) \cos(x) dx = (1 - \sin^2(x)) du = (1 - u^2) du$$. 6. **Substitute into the integral:** $$\int 8 \frac{\cos^3(x)}{\sin(x)} dx = \int 8 \frac{1 - u^2}{u} du = 8 \int \left(\frac{1}{u} - u\right) du$$ 7. **Integrate term by term:** $$8 \int \frac{1}{u} du - 8 \int u du = 8 \ln|u| - 8 \frac{u^2}{2} + C = 8 \ln|\sin(x)| - 4 \sin^2(x) + C$$ **Final answer:** $$\boxed{8 \ln|\sin(x)| - 4 \sin^2(x) + C}$$