https://file.aiquickdraw.com/mathbot/file/1725762849769nfarca2x.jpg

The integral shown in the image is:

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

To solve this, let's proceed step-by-step.

### Step 1: Factor the denominator
The denominator \( x^2 - 1 \) can be factored as:
\[
x^2 - 1 = (x - 1)(x + 1)
\]

### Step 2: Perform polynomial division
We divide \( \frac{x^3 + 1}{x^2 - 1} \). Let's carry out the division:

\[
\frac{x^3 + 1}{x^2 - 1} = x + \frac{2x}{x^2 - 1}
\]

### Step 3: Split the integral
Now we split the integral as follows:

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

### Step 4: Solve each part
- The first integral is straightforward:
  \[
  \int x \, dx = \frac{x^2}{2}
  \]

- The second integral requires substitution. Let \( u = x^2 - 1 \), so \( du = 2x \, dx \). The integral becomes:
  \[
  2 \int \frac{x}{x^2 - 1} \, dx = \int \frac{1}{u} \, du = \ln|u| = \ln|x^2 - 1|
  \]

### Step 5: Combine the results
The final solution is:
\[
\frac{x^2}{2} + \ln|x^2 - 1| + C
\]
where \( C \) is the constant of integration.

Let me know if you would like further details or have any questions! Here are a few follow-up questions for further learning:

1. How would you approach integrating more complex rational functions?
2. What are other methods for solving integrals involving polynomial divisions?
3. How does substitution simplify integrals with higher-order terms?
4. Can you apply partial fraction decomposition to a similar integral? How would it change the approach?
5. How do you determine the most appropriate integration technique based on the integrand?

**Tip**: Whenever dealing with rational functions, always check if polynomial division can simplify the expression before applying more advanced techniques like partial fractions!

Evaluate the integral of (x^3 + 1) / (x^2 - 1) dx.

Learn how to evaluate the integral of (x^3 + 1) / (x^2 - 1) using polynomial division and substitution. This calculus problem involves key concepts such as polynomial division and logarithmic integration, suitable for students in Grades 11-12.

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