1. **Stating the problem:** Evaluate the integral $$\int (\cos^{-1} 3x) \frac{x}{\sqrt{1 - 9x^2}} \, dx.$$\n\n2. **Understanding the integral:** The integrand is a product of the inverse cosine function and a rational expression involving $x$ and a square root. We will use substitution and integration by parts.\n\n3. **Substitution:** Let $$u = \cos^{-1} 3x.$$ Then, $$\cos u = 3x.$$ Differentiating both sides with respect to $x$, we get $$-\sin u \frac{du}{dx} = 3 \implies \frac{du}{dx} = -\frac{3}{\sin u}.$$\n\n4. **Expressing $x$ and $dx$ in terms of $u$:** From $$\cos u = 3x,$$ we have $$x = \frac{\cos u}{3}.$$ Also, $$\sin u = \sqrt{1 - \cos^2 u} = \sqrt{1 - 9x^2}.$$\n\n5. **Rewrite the integral:** Substitute $x$ and $dx$ in the integral. Note that $$\frac{x}{\sqrt{1 - 9x^2}} dx = \frac{\cos u / 3}{\sin u} dx.$$ From step 3, $$du = -\frac{3}{\sin u} dx \implies dx = -\frac{\sin u}{3} du.$$\n\n6. **Substitute $dx$ into the expression:**\n$$\frac{x}{\sqrt{1 - 9x^2}} dx = \frac{\cos u}{3 \sin u} \times \left(-\frac{\sin u}{3} du\right) = -\frac{\cos u}{9} du.$$\n\n7. **Integral becomes:**\n$$\int (\cos^{-1} 3x) \frac{x}{\sqrt{1 - 9x^2}} dx = \int u \left(-\frac{\cos u}{9}\right) du = -\frac{1}{9} \int u \cos u \, du.$$\n\n8. **Integration by parts:** Let $$I = \int u \cos u \, du.$$ Choose $$f = u, \quad dg = \cos u \, du,$$ then $$df = du, \quad g = \sin u.$$\n\n9. **Apply integration by parts formula:**\n$$I = u \sin u - \int \sin u \, du = u \sin u + \cos u + C.$$\n\n10. **Substitute back:**\n$$\int (\cos^{-1} 3x) \frac{x}{\sqrt{1 - 9x^2}} dx = -\frac{1}{9} (u \sin u + \cos u) + C = -\frac{1}{9} \left( (\cos^{-1} 3x) \sin (\cos^{-1} 3x) + \cos (\cos^{-1} 3x) \right) + C.$$\n\n11. **Simplify expressions:**\nSince $$\cos (\cos^{-1} 3x) = 3x,$$ and $$\sin (\cos^{-1} 3x) = \sqrt{1 - (3x)^2} = \sqrt{1 - 9x^2},$$\nwe get\n$$= -\frac{1}{9} \left( (\cos^{-1} 3x) \sqrt{1 - 9x^2} + 3x \right) + C.$$\n\n**Final answer:**\n$$\boxed{\int (\cos^{-1} 3x) \frac{x}{\sqrt{1 - 9x^2}} \, dx = -\frac{1}{9} \left( (\cos^{-1} 3x) \sqrt{1 - 9x^2} + 3x \right) + C}.$$