Definite Integration
Integration of exponential functions
Grade Class 12
Question:
If ∫ \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} dx = Ax + B \ln |9e^{2x} - 4| + C, then
(A) A + 18B = 16
(B) 18B - A = 19
(C) A - 18B = 17
(D) A + 18B = 32
Step-by-Step Solution
Key Concept: Multiply numerator and denominator by e^x to simplify the integrand.
Multiply by e^x/e^x: \int (4e^2x + 6) / (9e^2x - 4) dx. Let 9e^2x - 4 = t, then 18e^2x dx = dt. We can write 4e^2x + 6 = (4/9)(9e^2x - 4) + 16/9 + 6 = (4/9)(9e^2x - 4) + 70/9. Integral = \int (4/9 + (70/9) / (9e^2x - 4)) dx = 4x/9 + (70/9) \int dx / (9e^2x - 4). This leads to the form Ax + B ln |9e^2x - 4| + C.
Correct Answer: A,B