Thank you for the clarification. The integral is:

\[
\int_{8}^{9} \frac{12}{x^2 - 36} \, dx
\]

Let's solve this integral using partial fraction decomposition and evaluate it over the correct limits.

### Step 1: Factor the denominator
The denominator is factored as:

\[
x^2 - 36 = (x - 6)(x + 6)
\]

### Step 2: Partial Fraction Decomposition
We express \(\frac{12}{(x - 6)(x + 6)}\) as:

\[
\frac{12}{(x - 6)(x + 6)} = \frac{A}{x - 6} + \frac{B}{x + 6}
\]

Multiply both sides by \((x - 6)(x + 6)\):

\[
12 = A(x + 6) + B(x - 6)
\]

Expanding and simplifying:

\[
12 = A(x + 6) + B(x - 6) = Ax + 6A + Bx - 6B
\]

Combine like terms:

\[
12 = (A + B)x + (6A - 6B)
\]

This gives the system of equations:

1. \(A + B = 0\)
2. \(6A - 6B = 12\)

From the first equation, \(A = -B\). Substituting into the second equation:

\[
6(-B) - 6B = 12
\]
\[
-6B - 6B = 12
\]
\[
-12B = 12 \quad \Rightarrow \quad B = -1
\]

Since \(A = -B\), we have \(A = 1\).

Thus, the partial fraction decomposition is:

\[
\frac{12}{(x - 6)(x + 6)} = \frac{1}{x - 6} - \frac{1}{x + 6}
\]

### Step 3: Evaluate the Integral
Now, evaluate the integral:

\[
\int_{8}^{9} \left(\frac{1}{x - 6} - \frac{1}{x + 6}\right) dx
\]

This can be separated as:

\[
\int_{8}^{9} \frac{1}{x - 6} \, dx - \int_{8}^{9} \frac{1}{x + 6} \, dx
\]

Each integral is of the form \(\int \frac{1}{u} du = \ln |u|\), so:

\[
\int_{8}^{9} \frac{1}{x - 6} \, dx = \ln |x - 6| \Big|_{8}^{9} = \ln |9 - 6| - \ln |8 - 6| = \ln(3) - \ln(2)
\]

\[
\int_{8}^{9} \frac{1}{x + 6} \, dx = \ln |x + 6| \Big|_{8}^{9} = \ln |9 + 6| - \ln |8 + 6| = \ln(15) - \ln(14)
\]

So, the integral is:

\[
\left(\ln\left(\frac{3}{2}\right)\right) - \left(\ln\left(\frac{15}{14}\right)\right)
\]

This simplifies to:

\[
\ln\left(\frac{3}{2}\right) - \ln\left(\frac{15}{14}\right) = \ln\left(\frac{3}{2} \cdot \frac{14}{15}\right) = \ln\left(\frac{42}{30}\right) = \ln\left(\frac{7}{5}\right)
\]

Finally, compute this:

\[
\ln\left(\frac{7}{5}\right) \approx 0.336
\]

### Conclusion
The correct answer is approximately \(0.336\).

Would you like further details or have any other questions?

---

### Related Questions
1. How does changing the limits of integration affect the result?
2. What are other examples where partial fractions are used to simplify integrals?
3. What would be the result if the integral was evaluated from 6 to 8 instead?
4. How would you approach this problem if the denominator was not factorable easily?
5. What is the general rule for setting up partial fraction decomposition for more complex denominators?

### Tip:
Always simplify the expression inside the logarithm after integrating, as it can sometimes reduce to a simpler form.

Explore the solution to the integral of 9 to 8, specifically \(\int_{8}^{9} \frac{12}{x^2 - 36} \, dx\), using partial fraction decomposition and logarithmic functions. Understand how changing integration limits affects results and gain insights into applying partial fractions to simplify integrals.

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