If $\int\frac{(x-1)^{2}}{(x^{2}+1)^{2}}dx=tan^{-1}x+g(x)+c,$ then $g(x)=$

Question

Accepted Answer

To determine why Option 1 is the correct answer for the function $ g(x) $ in the integral $\int\frac{(x-1)^{2}}{(x^{2}+1)^{2}}dx=\tan^{-1}x+g(x)+c$, we need to analyze the integration process and the structure of the given integral.

### Explanation for Option 1:

1. **Integration by Parts**: The integral $\int\frac{(x-1)^{2}}{(x^{2}+1)^{2}}dx$ can be approached using integration by parts or substitution methods. The presence of $\tan^{-1}x$ in the result suggests that part of the integral involves the derivative of $\tan^{-1}x$, which is $\frac{1}{x^2+1}$.

2. **Decomposition**: The expression $(x-1)^2$ can be expanded to $x^2 - 2x + 1$. This allows us to decompose the integral into simpler parts:
   $$
   \int \frac{x^2}{(x^2+1)^2} \, dx - 2 \int \frac{x}{(x^2+1)^2} \, dx + \int \frac{1}{(x^2+1)^2} \, dx
   $$

3. **Recognizing Derivatives**: The derivative of $\tan^{-1}x$ is $\frac{1}{x^2+1}$, which is directly related to the integral of $\frac{1}{(x^2+1)^2}$. This suggests that the integral of $\frac{1}{(x^2+1)^2}$ contributes to the function $g(x)$.

4. **Matching Terms**: Given that $\tan^{-1}x$ is already part of the integral's result, the remaining part, $g(x)$, must account for the additional terms. The function $\frac{1}{x^2+1}$ is a logical candidate for $g(x)$ because it complements the $\tan^{-1}x$ term and fits the structure of the integral.

### Explanation for Why Other Options are Incorrect:

- **Option 2: $\tan^{-1}(\frac{x}{2})$**: This function does not naturally arise from the integration of $\frac{(x-1)^2}{(x^2+1)^2}$. The derivative of $\tan^{-1}(\frac{x}{2})$ is $\frac{1}{2+x^2}$, which does not match any part of the decomposition of the original integral.

- **Option 3: $\frac{1}{2(x^2+1)}$**: While this function is similar to $\frac{1}{x^2+1}$, it introduces an unnecessary factor of $\frac{1}{2}$, which does not align with the integral's structure or the given result.

- **Option 4: $\frac{1}{2(x+1)}$**: This function is unrelated to the integral's decomposition. The presence of $x+1$ in the denominator does not correspond to any part of the original integral's structure or the derivative of $\tan^{-1}x$.

In conclusion, Option 1, $\frac{1}{x^2+1}$, is the correct choice for $g(x)$ because it logically complements the $\tan^{-1}x$ term in the integral and fits the structure of the decomposition.