Find the antiderivative of the function.

We use the power rule for integration, $\int x^n \,dx = \frac{x^{n+1}}{n+1}$.

$$\int (3x^2 - 4x + 5) \,dx = 3\frac{x^{2+1}}{2+1} - 4\frac{x^{1+1}}{1+1} + 5x + C$$ $$= 3\frac{x^3}{3} - 4\frac{x^2}{2} + 5x + C$$ $$= x^3 - 2x^2 + 5x + C$$

For definite integrals, we don't need the constant $C$. So, the antiderivative is $F(x) = x^3 - 2x^2 + 5x$.

The definite integral is evaluated as $F(b) - F(a)$, where $b=3$ and $a=2$.

$$\int_{2}^{3} (3x^2 - 4x + 5) \,dx = \left[x^3 - 2x^2 + 5x\right]_{2}^{3}$$

$$F(3) = (3)^3 - 2(3)^2 + 5(3)$$ $$= 27 - 2(9) + 15$$ $$= 27 - 18 + 15$$ $$= 9 + 15 = 24$$

$$F(2) = (2)^3 - 2(2)^2 + 5(2)$$ $$= 8 - 2(4) + 10$$ $$= 8 - 8 + 10$$ $$= 0 + 10 = 10$$

$$F(3) - F(2) = 24 - 10$$ $$= 14$$

The final answer is $\text{14}$.