Question

Evaluate the integral \( \int \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} dx = Ax + B \log |9e^{2x} - 4| + c \), then (where \( c \) is the constant of integration)

• A. \( A = \frac{3}{2}, B = \frac{35}{36} \)
• B. \( A = \frac{1}{2}, B = \frac{35}{36} \)
• C. \( A = -\frac{3}{2}, B = \frac{35}{36} \)
• D. \( A = -\frac{3}{2}, B = \frac{36}{35} \)

Hint:

For integrals involving rational functions, use substitution methods to simplify and integrate.

Answer 1

The Correct Option is C

Step 1: Simplify the integral.
We are given the integral \( \int \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} dx \). Use substitution and properties of exponential functions to simplify the expression. Step 2: Apply integration.
After performing the integration using the appropriate substitution method, we find that: \[ A = -\frac{3}{2}, \quad B = \frac{35}{36} \] Step 3: Conclusion.
The correct answer is (C) \( A = -\frac{3}{2}, B = \frac{35}{36} \).