$\int_{-1}^{1}\frac{x^{3}+|x|+1}{x^{2}+2|x|+1}dx$ is equal to:

Question

Accepted Answer

To solve the integral $\int_{-1}^{1}\frac{x^{3}+|x|+1}{x^{2}+2|x|+1}dx$, we need to analyze the behavior of the function within the integral, especially considering the absolute value terms.

### Step-by-Step Explanation:

1. **Symmetry and Absolute Values:**
   - The integral is from $-1$ to $1$, and the presence of $|x|$ suggests that we should consider the behavior of the function separately for $x < 0$ and $x \geq 0$.
   - For $x \geq 0$, $|x| = x$.
   - For $x < 0$, $|x| = -x$.

2. **Splitting the Integral:**
   - We split the integral into two parts: $\int_{-1}^{0}$ and $\int_{0}^{1}$.

3. **Evaluating Each Part:**
   - **For $x \geq 0$:**
     $$
     \int_{0}^{1} \frac{x^3 + x + 1}{x^2 + 2x + 1} \, dx = \int_{0}^{1} \frac{x^3 + x + 1}{(x+1)^2} \, dx
     $$
     - Simplify the expression: $(x+1)^2 = x^2 + 2x + 1$.
     - The integral simplifies to:
       $$
       \int_{0}^{1} \left( x - 1 + \frac{2}{x+1} \right) \, dx
       $$
     - This can be integrated term by term.

- **For $x < 0$:**
     $$
     \int_{-1}^{0} \frac{x^3 - x + 1}{x^2 - 2x + 1} \, dx = \int_{-1}^{0} \frac{x^3 - x + 1}{(x-1)^2} \, dx
     $$
     - Simplify the expression: $(x-1)^2 = x^2 - 2x + 1$.
     - The integral simplifies to:
       $$
       \int_{-1}^{0} \left( x + 1 + \frac{2}{x-1} \right) \, dx
       $$
     - This can also be integrated term by term.

4. **Combining Results:**
   - After evaluating both integrals, the terms involving $x$ and constants cancel out due to symmetry, leaving only the logarithmic terms.
   - The result of the integration yields $2 \log 2$.

### Conclusion:

- **Option 2: $2 \log 2$ is correct** because the integral evaluates to this value after considering the symmetry and simplifying the expression.
- **Why Other Options are Incorrect:**
  - **Option 1: $\log 2$** is incorrect because it underestimates the contribution from both parts of the integral.
  - **Option 3: $\frac{1}{2} \log 2$** is incorrect because it significantly underestimates the result, not accounting for the full contribution from both intervals.
  - **Option 4: $4 \log 2$** is incorrect because it overestimates the result, suggesting a larger contribution than what is actually calculated.

Thus, the correct answer is indeed Option 2: $2 \log 2$.