The integral you've provided is written as:

$\int_0^3 (x - 2) \, dx$

Let's solve this step by step:

Step 1: Set up the integral

$\int_0^3 (x - 2) \, dx$

Step 2: Break the integral into two parts

We can split the integral as follows: $\int_0^3 x \, dx - \int_0^3 2 \, dx$

Step 3: Solve each part separately

Part 1: $\int_0^3 x \, dx$

Using the power rule for integrals: $\int x \, dx = \frac{x^2}{2}$ Now evaluate this from 0 to 3: $\left[\frac{x^2}{2}\right]_0^3 = \frac{3^2}{2} - \frac{0^2}{2} = \frac{9}{2}$

Part 2: $\int_0^3 2 \, dx$

This is a constant, so the integral of 2 is: $\int 2 \, dx = 2x$ Evaluate this from 0 to 3: $\left[2x\right]_0^3 = 2(3) - 2(0) = 6$

Step 4: Combine the results

Now subtract the two results: $\frac{9}{2} - 6 = \frac{9}{2} - \frac{12}{2} = \frac{-3}{2}$

So, the value of the integral is:

$\boxed{-\frac{3}{2}}$

Would you like further details or clarifications on this? Here are some related questions:

1. How would this integral change if the limits were different?
2. Can we visualize the area represented by this integral?
3. What is the geometric interpretation of negative integral results?
4. How would the result change if we integrated $x + 2$ instead of $x - 2$?
5. What are the fundamental rules for splitting an integral into simpler parts?

Tip: Always check if you can break down an integral into simpler terms for easier evaluation!