If `int(x-1)/(x^2sqrt(2x^2-2x-1))dx`= `sqrt(f(x))/g(x)`+c then the value of `f(x)` and `g(x)` is

Options

A. `f(x) = 2x^(2)-2x+1`
B. `g(x) = x+1`
C. `g(x)=x`
D. `f(x) =2x^(2)-2x`

Explanation

To solve the integral \( \int \frac{x-1}{x^2 \sqrt{2x^2 - 2x - 1}} \, dx \) and express it in the form \( \frac{\sqrt{f(x)}}{g(x)} + C \), we will follow these steps:

### Step 1: Simplify the Integral
We start with the integral:
\[ \int \frac{x-1}{x^2 \sqrt{2x^2 - 2x - 1}} \, dx \]
We can rewrite the integrand by factoring \( x^2 \) out of the square root in the denominator:
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