Step 1 : Understanding the Question:
The topic for this problem is Integral Calculus, specifically focusing on Special Algebraic Integrals. The integral $\int \frac{x^2+1}{x^4+1} dx$ is a classic problem that requires a specific algebraic manipulation to convert it into a standard trigonometric form. Direct substitution or simple partial fractions are difficult here because $x^4+1$ does not have real linear factors. Instead, we use a technique involving dividing by $x^2$.
Step 2 : Key Formulas and approach:
1. Algebraic Manipulation: Divide both the numerator and the denominator by $x^2$.
2. Substitution Trick: Let $t = x - \frac{1}{x}$. Then $dt = (1 + \frac{1}{x^2})dx$.
3. Completing the Square: Note that $(x - \frac{1}{x})^2 = x^2 + \frac{1}{x^2} - 2$.
4. Standard Integral: $\int \frac{dt}{t^2 + a^2} = \frac{1}{a} \tan^{-1}(\frac{t}{a}) + C$.
5. Approach: Divide by $x^2$, rewrite the denominator in terms of $(x - 1/x)$, and use substitution to solve.
Step 3 : Detailed Explanation:
We start by dividing both the numerator and the denominator by $x^2$: $I = \int \frac{1 + 1/x^2}{x^2 + 1/x^2} dx$.
The numerator $(1 + 1/x^2)$ is the derivative of $(x - 1/x)$. This suggests we should express the denominator in terms of $(x - 1/x)$.
Since $(x - 1/x)^2 = x^2 + 1/x^2 - 2$, we can say that $x^2 + 1/x^2 = (x - 1/x)^2 + 2$.
Now substitute these into the integral: $I = \int \frac{(1 + 1/x^2)}{(x - 1/x)^2 + 2} dx$.
Let $t = x - 1/x$. Then $dt = (1 + 1/x^2) dx$.
The integral becomes $I = \int \frac{dt}{t^2 + 2}$. We can rewrite 2 as $(\sqrt{2})^2$.
This is now in the standard form $\int \frac{dt}{t^2 + a^2}$ where $a = \sqrt{2}$.
Applying the integration formula: $I = \frac{1}{\sqrt{2}} \tan^{-1} (\frac{t}{\sqrt{2}}) + C$.
Finally, substitute back $t = x - 1/x$: $I = \frac{1}{\sqrt{2}} \tan^{-1} \left(\frac{x - 1/x}{\sqrt{2}}\right) + C$.
Simplify the expression inside the tangent inverse: $\frac{x - 1/x}{\sqrt{2}} = \frac{x^2 - 1}{\sqrt{2}x}$.
Thus, the final result is $\frac{1}{\sqrt{2}} \tan^{-1} \left(\frac{x^2 - 1}{\sqrt{2}x}\right) + C$.
Step 4 : Final Answer:
After algebraic manipulation and using the substitution $t = x - 1/x$, the integral evaluates to $\frac{1}{\sqrt{2}}\tan^{-1}(\frac{x^2-1}{\sqrt{2}x})+C$, which is Option (A).