Step 1: Understanding the Concept:
This integral involves a rational algebraic function with a square root in the denominator containing a fourth-degree polynomial.
Direct substitution like \( u = 2x^4 - 2x^2 + 1 \) would not work because the derivative would involve \( x^3 \), but we have a complex numerator.
The objective in such problems is to rearrange the terms so that the numerator becomes the derivative of the expression inside the radical.
A common technique for high-degree algebraic integrals is to "extract" the highest power of \( x \) from the radical.
This usually transforms the integrand into a form where a simple substitution solves the problem.
Step 2: Key Formula or Approach:
We manipulate the expression inside the square root by taking \( x^4 \) common:
\[ \sqrt{2x^4 - 2x^2 + 1} = \sqrt{x^4 \left( 2 - \frac{2}{x^2} + \frac{1}{x^4} \right)} = x^2 \sqrt{2 - \frac{2}{x^2} + \frac{1}{x^4}} \]
Then, we substitute this back into the integral and divide the numerator terms by the resulting power of \( x \).
Step 3: Detailed Explanation:
Substitute the modified radical back into the integral \( I \):
\[ I = \int \frac{x^2 - 1}{x^3 \cdot x^2 \sqrt{2 - \frac{2}{x^2} + \frac{1}{x^4}}} \, dx \]
Combine the powers of \( x \) in the denominator:
\[ I = \int \frac{x^2 - 1}{x^5 \sqrt{2 - \frac{2}{x^2} + \frac{1}{x^4}}} \, dx \]
Now, divide the terms in the numerator by \( x^5 \):
\[ I = \int \frac{\frac{x^2}{x^5} - \frac{1}{x^5}}{\sqrt{2 - \frac{2}{x^2} + \frac{1}{x^4}}} \, dx = \int \frac{\frac{1}{x^3} - \frac{1}{x^5}}{\sqrt{2 - \frac{2}{x^2} + \frac{1}{x^4}}} \, dx \]
Let us perform a substitution. Let:
\[ t = 2 - \frac{2}{x^2} + \frac{1}{x^4} = 2 - 2x^{-2} + x^{-4} \]
Differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = 0 - 2(-2)x^{-3} + (-4)x^{-5} = \frac{4}{x^3} - \frac{4}{x^5} \]
We can factor out 4:
\[ dt = 4 \left( \frac{1}{x^3} - \frac{1}{x^5} \right) \, dx \implies \left( \frac{1}{x^3} - \frac{1}{x^5} \right) \, dx = \frac{dt}{4} \]
Now, substitute these into the integral:
\[ I = \int \frac{1/4}{\sqrt{t}} \, dt = \frac{1}{4} \int t^{-1/2} \, dt \]
Integrating using the power rule \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \):
\[ I = \frac{1}{4} \left( \frac{t^{1/2}}{1/2} \right) + C = \frac{1}{4} \cdot 2\sqrt{t} + C = \frac{1}{2}\sqrt{t} + C \]
Finally, substitute the value of \( t \) back in terms of \( x \):
\[ I = \frac{1}{2} \sqrt{2 - \frac{2}{x^2} + \frac{1}{x^4}} + C \]
Step 4: Final Answer:
The result of the integration is \( \frac{1}{2} \sqrt{2 - \frac{2}{x^2} + \frac{1}{x^4}} + C \).
This is exactly option (C).