The given integral is:

$\int_{-1}^{1} \left( x^2 + 1 \right) \, dx$

Step 1: Separate the integral

This can be broken up into two simpler integrals:

$\int_{-1}^{1} \left( x^2 + 1 \right) \, dx = \int_{-1}^{1} x^2 \, dx + \int_{-1}^{1} 1 \, dx$

Step 2: Evaluate each integral

1. For the integral of $x^2$:

$\int_{-1}^{1} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{-1}^{1}$

Substitute the limits of integration:

$= \frac{1^3}{3} - \frac{(-1)^3}{3} = \frac{1}{3} - \left( -\frac{1}{3} \right) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}$

2. For the integral of $1$:

$\int_{-1}^{1} 1 \, dx = \left[ x \right]_{-1}^{1}$

Substitute the limits of integration:

$= 1 - (-1) = 1 + 1 = 2$

Step 3: Combine the results

Now, add the two results:

$\int_{-1}^{1} \left( x^2 + 1 \right) \, dx = \frac{2}{3} + 2 = \frac{2}{3} + \frac{6}{3} = \frac{8}{3}$

Final Answer:

$\int_{-1}^{1} \left( x^2 + 1 \right) \, dx = \frac{8}{3}$

Would you like further details or have any questions about this result?

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Here are 5 related questions that may interest you:

1. How do we compute definite integrals of odd and even functions?
2. What are the properties of integrals involving even functions over symmetric intervals?
3. How would this problem change if the limits of integration were different, say from 0 to 1?
4. What happens when you integrate a constant function over any interval?
5. How can you interpret the result of an integral geometrically?

Tip: When integrating polynomials, you can often use the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, valid for $n \neq -1$.