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The integral you have provided is:

\[
\int \frac{2 + V}{1 - V^2} \, dV
\]

We are asked to solve this using partial fractions. Let's break it down:

1. **Factor the denominator**:
   \[
   1 - V^2 = (1 - V)(1 + V)
   \]

2. **Set up partial fractions**: 
   We want to express the integrand as:
   \[
   \frac{2 + V}{(1 - V)(1 + V)} = \frac{A}{1 - V} + \frac{B}{1 + V}
   \]

   Multiply both sides by the denominator \((1 - V)(1 + V)\):
   \[
   2 + V = A(1 + V) + B(1 - V)
   \]

3. **Solve for \(A\) and \(B\)**:
   Expand the right-hand side:
   \[
   2 + V = A(1 + V) + B(1 - V) = A + AV + B - BV
   \]
   Collect like terms:
   \[
   2 + V = (A + B) + (A - B)V
   \]

   Now, equate the coefficients of like powers of \(V\):
   - For the constant term: \(A + B = 2\)
   - For the \(V\) term: \(A - B = 1\)

   Solve this system of equations:
   - Adding both equations: \(2A = 3 \implies A = \frac{3}{2}\)
   - Substituting into \(A + B = 2\): \(\frac{3}{2} + B = 2 \implies B = \frac{1}{2}\)

4. **Rewrite the integrand**:
   Now that we have \(A = \frac{3}{2}\) and \(B = \frac{1}{2}\), we can rewrite the integrand:
   \[
   \frac{2 + V}{(1 - V)(1 + V)} = \frac{3/2}{1 - V} + \frac{1/2}{1 + V}
   \]

5. **Integrate**:
   Now integrate each term:
   \[
   \int \left(\frac{3/2}{1 - V} + \frac{1/2}{1 + V}\right) dV = \frac{3}{2} \int \frac{1}{1 - V} dV + \frac{1}{2} \int \frac{1}{1 + V} dV
   \]

   These integrals are straightforward:
   \[
   \frac{3}{2} \ln|1 - V| - \frac{1}{2} \ln|1 + V| + C
   \]

6. **Simplify**:
   Using properties of logarithms, we can combine the terms:
   \[
   \frac{3}{2} \ln|1 - V| - \frac{1}{2} \ln|1 + V| = \ln \left( \frac{(1 - V)^{3/2}}{(1 + V)^{1/2}} \right)
   \]

   So the final solution is:
   \[
   \ln \left( \frac{(1 - V)^{3/2}}{(1 + V)^{1/2}} \right) + C
   \]

Let me know if you need further details or clarifications!

### Related Questions:
1. How do we handle partial fractions when the denominator has repeated roots?
2. What are the general steps for solving an integral by partial fractions?
3. Can partial fractions be used for improper fractions as well?
4. How does the logarithmic property help simplify expressions in integrals?
5. What are other common methods for solving rational integrals besides partial fractions?

### Tip:
When solving integrals by partial fractions, always check if the denominator can be factored first, as it simplifies the setup significantly.

∫(2 + V) / (1 - V^2) dV by partial fractions

This detailed solution explores how to solve the integral ∫(2 + V) / (1 - V^2) dV using partial fractions. Learn how to factor the denominator, decompose the rational function, and integrate step by step. Suitable for college-level calculus students.

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