If `f(x)=int(3x^2+1)/((x^2-1)^3)dxa n df(0)=0,` then the value of `|2/(f(2))|` is___

Explanation

To solve the problem, we need to evaluate the function \( f(x) \) defined as:

\[ f(x) = \int \frac{3x^2 + 1}{(x^2 - 1)^3} \, dx \]

Given that \( f(0) = 0 \), we need to find the value of \( \left| \frac{2}{f(2)} \right| \).

### Step 1: Rewrite the integrand

We can rewrite the numerator \( 3x^2 + 1 \) as follows:

\[ 3x^2 + 1 = 4x^2 - x^2 + 1 = 4x^2 - (x^2 - 1) \]

Thus, we have:

\[ f(x) = \int \frac{4x^2 - (x^2 - 1)}{(x^2 - 1)^3} \, dx = \int \left( \frac{4x^2}{(x^2 - 1)^3} - \frac{1}{(x^2 - 1)^3} \right) \, dx \]

### Step 2: Split the integral

Now we can split the integral into two parts:

\[ f(x) = \int \frac{4x^2}{(x^2 - 1)^3} \, dx - \int \frac{1}{(x^2 - 1)^3} \, dx \]

### Step 3: Evaluate the first integral using integration by parts

Let:

- \( u = x \) and \( dv = \frac{4x}{(x^2 - 1)^3} \, dx \)

Then:

- \( du = dx \)
- \( v = -\frac{2}{(x^2 - 1)^2} \) (after integrating)

Using integration by parts:

\[ \int u \, dv = uv - \int v \, du \]

We get:

\[ \int \frac{4x^2}{(x^2 - 1)^3} \, dx = -\frac{2x}{(x^2 - 1)^2} + \int \frac{2}{(x^2 - 1)^2} \, dx \]

### Step 4: Evaluate the second integral

The second integral \( \int \frac{1}{(x^2 - 1)^3} \, dx \) can also be evaluated using a standard integral formula or further integration techniques.

### Step 5: Combine results

After evaluating both integrals, we combine the results to express \( f(x) \).

### Step 6: Find \( f(2) \)

Now we will substitute \( x = 2 \) into our expression for \( f(x) \):

\[ f(2) = -\frac{2 \cdot 2}{(2^2 - 1)^2} + \text{(result from second integral)} \]

Calculating \( f(2) \):

\[ f(2) = -\frac{4}{(4 - 1)^2} + \text{(result from second integral)} = -\frac{4}{9} + \text{(result from second integral)} \]

### Step 7: Calculate \( |2/f(2)| \)

Finally, we compute:

\[ \left| \frac{2}{f(2)} \right| = \left| \frac{2}{-\frac{4}{9} + \text{(result from second integral)}} \right| \]