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The result of the definite integral 

\[
\int_{-1}^{1} \left[ 3\left(\frac{x+1}{x+2}\right) - (x+1) \right] \, dx
\]

is \( 4 - 3 \log(3) \).

Let me know if you'd like further details or explanations! Here are some questions related to this:

1. How do you approach integrating a rational function like \(\frac{x+1}{x+2}\)?
2. What is the significance of the natural logarithm \(\log(3)\) in the result?
3. How does the choice of limits of integration \([-1, 1]\) impact the calculation?
4. What would change if the integrand was multiplied by a different constant factor?
5. Can this result be interpreted in terms of areas under the curve?

**Tip:** When integrating rational functions, partial fraction decomposition can often simplify the process.

Help me integrate the expression \[ \int_{-1}^{1} \left[ 3\left(\frac{x+1}{x+2}\right) - (x+1) \right] dx \]

Learn how to evaluate the definite integral \( \int_{-1}^{1} \left[ 3\left(\frac{x+1}{x+2}\right) - (x+1) \right] dx \) with detailed steps. This solution involves concepts of rational functions and logarithmic integration, providing an answer of \( 4 - 3 \log(3) \). Suitable for advanced high school or college-level calculus students.

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