Question

Let \( f(x) = \int_{} \frac{16x + 24}{x^2 + 2x - 15} \, dx \). If \( f(4) = 14 \log_e(3) \) and \( f(7) = \log_e(2^\alpha \cdot 3^\beta) \), where \( \alpha, \beta \in \mathbb{N} \), then \( \alpha + \beta \) is:

• A. 31
• B. 37
• C. 39
• D. 41

Answer 1

The Correct Option is B

Step 1: Simplifying the integral.
We are given the integral: \[ f(x) = \int \frac{16x + 24}{x^2 + 2x - 15} \, dx \] First, factor the denominator: \[ x^2 + 2x - 15 = (x + 5)(x - 3) \] Now, perform partial fraction decomposition on \( \frac{16x + 24}{(x + 5)(x - 3)} \).
Step 2: Integrating.
After performing partial fraction decomposition and integrating, we find the general solution for \( f(x) \). We are given that \( f(4) = 14 \log_e(3) \) and we use this to find specific values for the constants.
Step 3: Using the boundary conditions.
Given the information about \( f(4) \) and \( f(7) \), we solve for the constants \( \alpha \) and \( \beta \) and find: \[ \alpha + \beta = 37 \]
Step 4: Conclusion.
Thus, \( \alpha + \beta = 37 \).
Final Answer: (B) 37