1. **State the problem:** Evaluate the definite integral of the function $6x^2 + 5$ from $x=1$ to $x=3$. 2. **Formula and rules:** The definite integral of a function $f(x)$ from $a$ to $b$ is given by $$\int_a^b f(x)\,dx = F(b) - F(a)$$ where $F(x)$ is the antiderivative of $f(x)$. 3. **Find the antiderivative:** For $f(x) = 6x^2 + 5$, the antiderivative is $$F(x) = \int (6x^2 + 5)\,dx = 2x^3 + 5x + C$$ where $C$ is the constant of integration. 4. **Evaluate the definite integral:** $$\int_1^3 (6x^2 + 5)\,dx = F(3) - F(1) = (2(3)^3 + 5(3) + C) - (2(1)^3 + 5(1) + C)$$ 5. **Simplify:** $$= (2 \times 27 + 15 + C) - (2 \times 1 + 5 + C)$$ $$= (54 + 15 + C) - (2 + 5 + C)$$ $$= 69 + C - 7 - C$$ 6. **Cancel constants:** $$= 69 - 7 = 62$$ **Final answer:** $$\boxed{62}$$