`int_(1)^(2)((x^(2)-1)dx)/(x^(3).sqrt(2x^(4)-2x^(2)+1))=(u)/(v)` where u and v are in their lowest form. Find the value of `((1000)u)/(v)`.

Options

A. 100
B. 125
C. 120
D. None of these

Answer

To solve the integral

\[ \int_{1}^{2} \frac{x^{2}-1}{x^{3} \sqrt{2x^{4}-2x^{2}+1}} \, dx, \]

we will follow these steps:

### Step 1: Simplify the Integral
First, we rewrite the integral by factoring out the highest power of \(x\) from the denominator's square root.

\[ \sqrt{2x^{4}-2x^{2}+1} = \sqrt{x^{4}(2 - \frac{2}{x^{2}} + \frac{1}{x^{4}})} = x^{2}\sqrt{2 - \frac{2}{x^{2}} + \frac{1}{x^{4}}}. \]

Thus, the integral becomes:

\[ \int_{1}^{2} \frac{x^{2}-1}{x^{3} \cdot x^{2} \sqrt{2 - \frac{2}{x^{2}} + \frac{1}{x^{4}}}} \, dx = \int_{1}^{2} \frac{x^{2}-1}{x^{5} \sqrt{2 - \frac{2}{x^{2}} + \frac{1}{x^{4}}}} \, dx. \]

### Step 2: Substitute Variables
To simplify the integral further, we can use a substitution. Let:

\[ t = 2 - \frac{2}{x^{2}} + \frac{1}{x^{4}}. \]

Calculating \(dt\):

\[ dt = \left( \frac{4}{x^{3}} - \frac{4}{x^{5}} \right) dx = \frac{4(x^{2}-1)}{x^{5}} dx. \]

Thus,

\[ dx = \frac{x^{5}}{4(x^{2}-1)} dt. \]

### Step 3: Change the Limits
When \(x = 1\):

\[ t = 2 - 2 + 1 = 1. \]

When \(x = 2\):

\[ t = 2 - \frac{2}{4} + \frac{1}{16} = 2 - 0.5 + 0.0625 = 1.5625 = \frac{25}{16}. \]

### Step 4: Rewrite the Integral
Now substituting everything into the integral, we have:

\[ \int_{1}^{\frac{25}{16}} \frac{1}{\sqrt{t}} \cdot \frac{1}{4} dt = \frac{1}{4} \int_{1}^{\frac{25}{16}} t^{-1/2} dt. \]

### Step 5: Evaluate the Integral
The integral of \(t^{-1/2}\) is:

\[ \int t^{-1/2} dt = 2t^{1/2}. \]

Thus, we evaluate:

\[ \frac{1}{4} \left[ 2t^{1/2} \right]_{1}^{\frac{25}{16}} = \frac{1}{4} \left[ 2 \left( \frac{5}{4} - 1 \right) \right] = \frac{1}{4} \left[ 2 \left( \frac{5}{4} - \frac{4}{4} \right) \right] = \frac{1}{4} \left[ 2 \cdot \frac{1}{4} \right] = \frac{1}{8}. \]

### Step 6: Express in Terms of u and v
The integral evaluates to:

\[ \frac{1}{8} = \frac{u}{v}. \]

Here, \(u = 1\) and \(v = 8\) in their lowest form.

### Step 7: Find the Value of \(\frac{1000u}{v}\)
Now we compute:

\[ \frac{1000u}{v} = \frac{1000 \cdot 1}{8} = 125. \]

Thus, the final answer is:

\[ \boxed{125}. \]