https://file.aiquickdraw.com/mathbot/file/172636541010937gix5xo.jpg

The integral in the image is:

\[
\int \frac{x^5 + 8x^3 + 7x}{(x^2 - 1)^2} \, dx
\]

Let's solve this integral step by step.

### Step 1: Simplify the integrand
The numerator is \( x^5 + 8x^3 + 7x \), which can be factored as \( x(x^4 + 8x^2 + 7) \), but this doesn't seem directly helpful. Instead, we will aim for a different strategy using partial fraction decomposition.

We recognize that the denominator is \( (x^2 - 1)^2 \), which suggests we may need to rewrite the numerator in a way that allows us to use this denominator structure.

### Step 2: Set up for partial fraction decomposition
Given the form of the denominator \( (x^2 - 1)^2 \), we decompose the fraction as follows:

\[
\frac{x^5 + 8x^3 + 7x}{(x^2 - 1)^2} = \frac{Ax + B}{x^2 - 1} + \frac{Cx + D}{(x^2 - 1)^2}
\]

We will solve for \( A, B, C, D \) and then integrate term by term.

Let me compute these coefficients and continue solving the integral.The integral simplifies to the following expression:

\[
\frac{(x^2 - 1)(x^2 + 10 \log(x^2 - 1)) - 16}{2(x^2 - 1)}
\]

This is the antiderivative of the given integral. You can now evaluate it further by plugging in specific bounds (if it's a definite integral) or simplify the result.

Would you like further details on the solution steps, or do you have any specific questions?

Here are 5 related questions:
1. How can you use partial fraction decomposition to solve other rational integrals?
2. What is the role of logarithmic functions in integration?
3. Can you apply the same method to an integral with different polynomial powers?
4. How would you handle this integral if it were a definite integral (with bounds)?
5. What types of integrals require partial fraction decomposition?

**Tip**: When dealing with complex rational functions in integration, partial fractions are often an efficient way to break down the problem.

Learn how to solve the integral of (x^5 + 8x^3 + 7x) / (x^2 - 1)^2 using partial fraction decomposition and logarithmic integration. This problem explores rational functions, simplifying expressions, and solving integrals with detailed steps. Suitable for college-level calculus students.

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