If ∫\frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} dx = Ax + B \ln |9e^{2x} - 4| + C, then
Step-by-Step Solution
Key Concept: Multiply numerator and denominator by e^x to transform the integral into a form where the denominator's derivative is related to the numerator, then use partial fractions or substitution.
Let I = \int(4e^x + 6e^-x)/(9e^x - 4e^-x) dx. Multiply numerator and denominator by e^x: I = \int(4e^2x + 6)/(9e^2x - 4) dx. Let u = e^2x, du = 2e^2x dx, so dx = du/(2u). The integral becomes \int(4u + 6)/(9u - 4) * (du/2u) = \int(2u + 3)/(u(9u - 4)) du. Using partial fractions: (2u + 3)/(u(9u - 4)) = A'/u + B'/(9u - 4). Solving gives A' = -3/4 and B' = 25/4. Integrating gives the form Ax + B ln |9e^2x - 4| + C. Comparing coefficients yields A and B.
Correct Answer: 1, 2