Integrate the following functions w.r.t x: (4e^x - 25)/(2e^x - 5)

Question

Integrate the following functions w.r.t x:

\(\frac{4e^x - 25}{2e^x - 5}\)

(4e^x - 25)/(2e^x - 5)

Answer 1

(4e^x - 25)/(2e^x - 5)

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

∴ 4ex – 25 = A(2ex – 5) + B [d/dx (2ex – 5)]

= A(2ex – 5) + B(2ex – 0) 

∴ 4ex – 25 = (2A + 2B) ex – 5A

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

2A + 2B = 4 …….(1) 

and 5A = 25 

∴ A = 5 from (1), 2(5) + 2B = 4 

∴ 2B = -6

∴ B = -3