Integrate the following functions w.r.t x: (3e^2x + 5)/(4e^2x​​​​​​​ - 5)

Question

Integrate the following functions w.r.t x:

\(\frac{3e^{2x} + 5}{4e^{2x} - 5}\)

(3e^2x + 5)/(4e^2x - 5)

Answer 1

(3e^2x + 5)/(4e^2x - 5)

Put, Numerator = A (Denominator) + B [d/dx (Denominator)]

∴ 3e2x + 5 = A(4e2x – 5) + B [d/dx (4e2x – 5)]

= A(4e2x – 5) + B(4 . e2x × 2 – 0) 

∴ 3e2x + 5 = (4A + 8B) e2x – 5A

Equating the coefficient of e2x and constant on both sides, we get

4A + 8B = 3 …….. (1) 

and -5A = 5 

∴ A = -1 

∴ from (1), 4(-1) + 8B = 3 

∴ 8B = 7

∴ B = 7/8