1. The problem is to evaluate the definite integral $$\int_5^8 \sin(x)\,dx$$. 2. Recall that the antiderivative of $\sin(x)$ is $-\cos(x)$. 3. Using the Fundamental Theorem of Calculus, we have: $$\int_5^8 \sin(x)\,dx = [-\cos(x)]_5^8 = -\cos(8) + \cos(5)$$ 4. This means we evaluate $\cos(x)$ at the upper limit 8 and the lower limit 5, then subtract. 5. So the exact value of the integral is: $$\cos(5) - \cos(8)$$ 6. If a decimal approximation is needed, use a calculator to find $\cos(5) \approx 0.2837$ and $\cos(8) \approx -0.1455$, so: $$0.2837 - (-0.1455) = 0.4292$$ 7. Therefore, the value of the integral is approximately $0.4292$.