1. **State the problem:** We need to evaluate the integral $$I = \int \frac{3x}{x^2 - 4x - 5} \, dx.$$\n\n2. **Identify the formula and approach:** The integral involves a rational function where the numerator is a linear polynomial and the denominator is a quadratic polynomial. A common method is to use partial fraction decomposition after factoring the denominator.\n\n3. **Factor the denominator:**\n$$x^2 - 4x - 5 = (x - 5)(x + 1).$$\n\n4. **Set up partial fractions:**\n$$\frac{3x}{(x - 5)(x + 1)} = \frac{A}{x - 5} + \frac{B}{x + 1}.$$\nMultiply both sides by the denominator to get:\n$$3x = A(x + 1) + B(x - 5).$$\n\n5. **Find coefficients A and B:**\nExpand the right side:\n$$3x = A x + A + B x - 5 B = (A + B) x + (A - 5 B).$$\nEquate coefficients of like terms:\n- Coefficient of $x$: $3 = A + B$\n- Constant term: $0 = A - 5 B$\n\n6. **Solve the system:**\nFrom $0 = A - 5 B$, we get $A = 5 B$. Substitute into $3 = A + B$:\n$$3 = 5 B + B = 6 B \implies B = \frac{3}{6} = \frac{1}{2}.$$\nThen, $A = 5 \times \frac{1}{2} = \frac{5}{2}.$\n\n7. **Rewrite the integral:**\n$$I = \int \left( \frac{5/2}{x - 5} + \frac{1/2}{x + 1} \right) dx = \frac{5}{2} \int \frac{1}{x - 5} dx + \frac{1}{2} \int \frac{1}{x + 1} dx.$$\n\n8. **Integrate:**\n$$I = \frac{5}{2} \ln|x - 5| + \frac{1}{2} \ln|x + 1| + C,$$\nwhere $C$ is the constant of integration.\n\n**Final answer:**\n$$\boxed{I = \frac{5}{2} \ln|x - 5| + \frac{1}{2} \ln|x + 1| + C}.$$