To solve the integral

$\int_{-2}^{2} \frac{1 + x^2}{1 + 2x} \, dx,$

we can use symmetry properties and substitution if necessary. Let's examine the function inside the integral for possible simplifications.

First, note that the integrand $\frac{1 + x^2}{1 + 2x}$ is not an even or odd function, which means we can't directly simplify it using symmetry around the y-axis.

However, we can consider a substitution to simplify the integration process. Let's try $u = 1 + 2x$, which means $du = 2dx$, or $dx = \frac{du}{2}$.

When $x = -2$, $u = 1 + 2(-2) = -3$.

When $x = 2$, $u = 1 + 2(2) = 5$.

Therefore, the limits of integration change from $[-2, 2]$ to $[-3, 5]$.

Now, the integral becomes:

$\int_{-3}^{5} \frac{1 + \left( \frac{u-1}{2} \right)^2}{u} \cdot \frac{1}{2} \, du.$

Simplifying inside the integrand:

$\left( \frac{u-1}{2} \right)^2 = \frac{(u-1)^2}{4} = \frac{u^2 - 2u + 1}{4}.$

Thus,

$1 + \left( \frac{u-1}{2} \right)^2 = 1 + \frac{u^2 - 2u + 1}{4} = \frac{4 + u^2 - 2u + 1}{4} = \frac{u^2 - 2u + 5}{4}.$

So, the integral becomes:

$\int_{-3}^{5} \frac{\frac{u^2 - 2u + 5}{4}}{u} \cdot \frac{1}{2} \, du = \frac{1}{8} \int_{-3}^{5} \frac{u^2 - 2u + 5}{u} \, du = \frac{1}{8} \int_{-3}^{5} \left( u - 2 + \frac{5}{u} \right) \, du.$

Breaking it into simpler integrals:

$\frac{1}{8} \left( \int_{-3}^{5} u \, du - 2 \int_{-3}^{5} du + 5 \int_{-3}^{5} \frac{1}{u} \, du \right).$

Solving each integral separately:

1. $\int_{-3}^{5} u \, du = \left[ \frac{u^2}{2} \right]_{-3}^{5} = \frac{25}{2} - \frac{9}{2} = \frac{16}{2} = 8.$
2. $-2 \int_{-3}^{5} du = -2[u]_{-3}^{5} = -2(5 - (-3)) = -2(8) = -16.$
3. $5 \int_{-3}^{5} \frac{1}{u} \, du = 5[\ln|u|]_{-3}^{5} = 5(\ln 5 - \ln 3) = 5 \ln \frac{5}{3}.$

Combining these results:

$\frac{1}{8} \left( 8 - 16 + 5 \ln \frac{5}{3} \right) = \frac{1}{8} \left( -8 + 5 \ln \frac{5}{3} \right) = -1 + \frac{5}{8} \ln \frac{5}{3}.$

Therefore, the value of the integral is:

$\int_{-2}^{2} \frac{1 + x^2}{1 + 2x} \, dx = -1 + \frac{5}{8} \ln \frac{5}{3}.$

Would you like further details on any step or have any other questions? Here are 5 more questions you can ask:

1. How do you determine if a function is even or odd?
2. What are common techniques for evaluating definite integrals?
3. Can you explain the process of u-substitution in integration?
4. How do you handle improper integrals?
5. What are some applications of definite integrals in real life?

Tip: Always double-check the limits of integration when performing a substitution to ensure accuracy.