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We are given that:

\[
\int_0^3 f(x) \, dx = -5
\]

We need to find the value of:

\[
\int_0^3 \left( f(x) + x - 1 \right) dx
\]

Using the linearity property of integrals, we can split the integral into three parts:

\[
\int_0^3 \left( f(x) + x - 1 \right) dx = \int_0^3 f(x) \, dx + \int_0^3 x \, dx - \int_0^3 1 \, dx
\]

Now we can compute each part:

1. The first part is already given:
   \[
   \int_0^3 f(x) \, dx = -5
   \]

2. The second part involves the integral of \(x\) from 0 to 3:
   \[
   \int_0^3 x \, dx = \left[ \frac{x^2}{2} \right]_0^3 = \frac{3^2}{2} - 0 = \frac{9}{2}
   \]

3. The third part is the integral of 1 from 0 to 3:
   \[
   \int_0^3 1 \, dx = \left[ x \right]_0^3 = 3 - 0 = 3
   \]

Now, summing up all three parts:

\[
\int_0^3 \left( f(x) + x - 1 \right) dx = -5 + \frac{9}{2} - 3
\]

Simplifying:

\[
-5 + \frac{9}{2} - 3 = -8 + \frac{9}{2} = \frac{-16}{2} + \frac{9}{2} = \frac{-7}{2}
\]

Thus, the value of the integral is:

\[
\boxed{-\frac{7}{2}}
\]

Do you need any further details or clarifications?

Here are some related questions for further exploration:
1. How would the result change if \( f(x) \) had different boundary values?
2. Can you apply this approach to other composite functions involving \( f(x) \)?
3. What are the steps for finding definite integrals of more complex functions?
4. How does the linearity of integration help in simplifying integrals like this one?
5. How can you interpret definite integrals geometrically?

**Tip:** Always apply the linearity of integration to break down complex expressions into simpler, computable parts!

Given that ∫[0 to 3] f(x) dx = -5, find the value of ∫[0 to 3] (f(x) + x - 1) dx.

Learn how to solve the integral ∫[0 to 3] (f(x) + x - 1) dx using the property of linearity of integration. This problem includes calculus concepts such as definite integrals and breaking them down step-by-step. Suitable for students in grades 11-12 or early college.

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