To solve the integral \(\int \frac{x}{(x^2-1)^2} \, dx\), we can use the substitution method. Let's go through the steps:

1. **Substitution**:
   Let \( u = x^2 - 1 \). Then, \( du = 2x \, dx \) or \( \frac{du}{2} = x \, dx \).

2. **Rewrite the integral**:
   Substituting \( u \) and \( du \) into the integral:
   \[
   \int \frac{x}{(x^2-1)^2} \, dx = \int \frac{x}{u^2} \cdot \frac{du}{2x}
   \]
   The \( x \) terms cancel out:
   \[
   \int \frac{1}{2u^2} \, du
   \]

3. **Integrate**:
   The integral now is:
   \[
   \int \frac{1}{2u^2} \, du = \frac{1}{2} \int u^{-2} \, du
   \]
   Using the power rule for integration \(\int u^n \, du = \frac{u^{n+1}}{n+1} + C\), we get:
   \[
   \frac{1}{2} \int u^{-2} \, du = \frac{1}{2} \left( \frac{u^{-1}}{-1} \right) + C = -\frac{1}{2u} + C
   \]

4. **Substitute back**:
   Replace \( u \) with \( x^2 - 1 \):
   \[
   -\frac{1}{2u} + C = -\frac{1}{2(x^2 - 1)} + C
   \]

So, the solution to the integral is:
\[
\int \frac{x}{(x^2-1)^2} \, dx = -\frac{1}{2(x^2 - 1)} + C
\]

Would you like any further details or have any questions?

Here are 5 questions you might want to ask next:
1. How do you solve a definite integral for a similar function?
2. Can you explain the substitution method in more detail?
3. How do you handle integrals with different forms of rational functions?
4. What are some common techniques for integrating rational functions?
5. Can you provide examples of more complex integrals and their solutions?

Tip: Always check if substitution or partial fraction decomposition can simplify the integration process.

Explore how to solve the integral of x over (x^2 - 1)^2 with a detailed step-by-step solution. This problem involves integration by substitution and is suitable for advanced high school students and beyond.

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