1. **State the problem:** We need to evaluate the integral $$\int \frac{5}{13 - 6x + x^2} \, dx.$$\n\n2. **Rewrite the denominator:** The quadratic in the denominator is $$x^2 - 6x + 13.$$ To integrate, we complete the square:\n$$x^2 - 6x + 13 = (x^2 - 6x + 9) + 4 = (x - 3)^2 + 4.$$\n\n3. **Rewrite the integral:**\n$$\int \frac{5}{(x - 3)^2 + 4} \, dx.$$\n\n4. **Use the standard integral formula:**\nFor $$\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C,$$\nwe identify $$a^2 = 4 \Rightarrow a = 2.$$\n\n5. **Apply the constant multiple:**\n$$\int \frac{5}{(x - 3)^2 + 2^2} \, dx = 5 \int \frac{1}{(x - 3)^2 + 2^2} \, dx = 5 \cdot \frac{1}{2} \arctan\left(\frac{x - 3}{2}\right) + C.$$\n\n6. **Simplify the result:**\n$$\frac{5}{2} \arctan\left(\frac{x - 3}{2}\right) + C.$$\n\n**Final answer:**\n$$\int \frac{5}{13 - 6x + x^2} \, dx = \frac{5}{2} \arctan\left(\frac{x - 3}{2}\right) + C.$$