The problem presented in the image is:

$\int_4^6 \frac{2}{x^2 - 4} \, dx = \ln(p)$

We are tasked with finding the value of $p$.

Step 1: Simplify the Integrand

We first factor the denominator: $x^2 - 4 = (x - 2)(x + 2)$ Thus, the integral becomes: $\int_4^6 \frac{2}{(x - 2)(x + 2)} \, dx$

Step 2: Apply Partial Fraction Decomposition

We decompose the fraction: $\frac{2}{(x - 2)(x + 2)} = \frac{A}{x - 2} + \frac{B}{x + 2}$ Multiplying both sides by $(x - 2)(x + 2)$, we get: $2 = A(x + 2) + B(x - 2)$ Expanding: $2 = A x + 2A + B x - 2B$ Group terms: $2 = (A + B) x + (2A - 2B)$ Now, solve the system of equations: $A + B = 0 \quad \text{(1)}$ $2A - 2B = 2 \quad \text{(2)}$

From equation (1), $A = -B$. Substitute into equation (2): $2(-B) - 2B = 2$ $-4B = 2$ $B = -\frac{1}{2}$ Thus, $A = \frac{1}{2}$.

Step 3: Rewrite the Integral

Now that we have the partial fractions, the integral becomes: $\int_4^6 \frac{1/2}{x - 2} \, dx - \int_4^6 \frac{1/2}{x + 2} \, dx$

This can be rewritten as: $\frac{1}{2} \left( \int_4^6 \frac{1}{x - 2} \, dx - \int_4^6 \frac{1}{x + 2} \, dx \right)$

Step 4: Evaluate the Integrals

The integrals are standard logarithmic integrals: $\int \frac{1}{x - a} \, dx = \ln|x - a|$ Thus, we have: $\frac{1}{2} \left( \ln|x - 2| \bigg|_4^6 - \ln|x + 2| \bigg|_4^6 \right)$

Now, calculate the values: $\ln|6 - 2| = \ln(4), \quad \ln|4 - 2| = \ln(2)$ $\ln|6 + 2| = \ln(8), \quad \ln|4 + 2| = \ln(6)$

Substitute these back: $\frac{1}{2} \left( (\ln(4) - \ln(2)) - (\ln(8) - \ln(6)) \right)$ Simplify: $\frac{1}{2} \left( \ln\left(\frac{4}{2}\right) - \ln\left(\frac{8}{6}\right) \right)$ $\frac{1}{2} \left( \ln(2) - \ln\left(\frac{4}{3}\right) \right)$ $\frac{1}{2} \ln\left(\frac{2}{4/3}\right) = \frac{1}{2} \ln\left(\frac{6}{4}\right) = \frac{1}{2} \ln\left(\frac{3}{2}\right)$

Step 5: Relate to $\ln(p)$

We know: $\frac{1}{2} \ln\left(\frac{3}{2}\right) = \ln(p)$ Thus: $p = \left(\frac{3}{2}\right)^{1/2} = \sqrt{\frac{3}{2}}$

Hence, the value of $p$ is: $p = \sqrt{\frac{3}{2}}$

Would you like more details on any of the steps? Here are some related questions to deepen your understanding:

1. How do you perform partial fraction decomposition for more complex expressions?
2. What is the importance of logarithmic properties in integrals?
3. How can we generalize this method for different integrals of rational functions?
4. What are some other methods for solving rational function integrals?
5. How does the behavior of the logarithmic function influence the integral's bounds?

Tip: When solving integrals of rational functions, partial fraction decomposition is a powerful tool, especially when dealing with quadratic denominators.