The given integral is:

\[
\int \frac{v^2}{1 - v^2} \, dv
\]

We can solve this using partial fractions. Let's proceed step by step:

### Step 1: Express the integrand in a simpler form
We start by rewriting the integrand:

\[
\frac{v^2}{1 - v^2} = \frac{v^2}{(1 - v)(1 + v)}
\]

We can rewrite \(v^2 = (1 - (1 - v^2))\), so:

\[
\frac{v^2}{1 - v^2} = 1 - \frac{1}{1 - v^2}
\]

Now the integral becomes:

\[
\int \left( 1 - \frac{1}{1 - v^2} \right) dv
\]

### Step 2: Break it up into two integrals
\[
\int 1 \, dv - \int \frac{1}{1 - v^2} \, dv
\]

The first integral is straightforward:

\[
\int 1 \, dv = v
\]

For the second integral, we use partial fractions. We know:

\[
\frac{1}{1 - v^2} = \frac{1}{2} \left( \frac{1}{1 - v} + \frac{1}{1 + v} \right)
\]

So the second integral becomes:

\[
\int \frac{1}{1 - v^2} \, dv = \frac{1}{2} \int \left( \frac{1}{1 - v} + \frac{1}{1 + v} \right) dv
\]

### Step 3: Solve each term
Now, solve both terms separately:

\[
\frac{1}{2} \int \frac{1}{1 - v} \, dv = -\frac{1}{2} \ln |1 - v|
\]

\[
\frac{1}{2} \int \frac{1}{1 + v} \, dv = \frac{1}{2} \ln |1 + v|
\]

### Step 4: Combine the results
Now, combine all terms:

\[
v - \frac{1}{2} \ln |1 - v| + \frac{1}{2} \ln |1 + v| + C
\]

Finally, simplify the logarithmic terms:

\[
v + \frac{1}{2} \ln \left( \frac{1 + v}{1 - v} \right) + C
\]

### Final Answer:
The integral is:

\[
\int \frac{v^2}{1 - v^2} \, dv = v + \frac{1}{2} \ln \left( \frac{1 + v}{1 - v} \right) + C
\]

Would you like further clarification or additional details on any step?

### Related Questions:
1. How do partial fraction decompositions work in integrals?
2. How can we handle integrals involving logarithmic functions?
3. Why do we need absolute value signs when integrating rational functions?
4. What are the common techniques for integrating rational functions?
5. How do we interpret the constant of integration?

### Tip:
Whenever you encounter rational functions, always consider partial fraction decomposition if the degree of the numerator is less than the degree of the denominator.

Learn how to solve the integral ∫ v^2 / (1 - v^2) dv using partial fraction decomposition. This step-by-step guide explains how to break down the rational function and solve using logarithmic integration. Ideal for high school seniors or college students studying Calculus I.

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