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Question
Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`
Solution
Let I = `int (20 + 12e^x)/(3e^x + 4).dx`
∴ 20 + 12ex = `"A"(3e^x + 4) + "B"d/dx(3e^x + 4)`
= 3Aex + 4A + 3Bex
∴ 20 + 12ex = 4A + (3A + 3B) ex
By Equating the coefficient of on both sides, we get
4A = 20 and 3A + 3B = 12
Solving these equations, we get
A = 5 and B = - 1
∴ 20 + 12ex = 5(3ex + 4) – 3ex
∴ I = `int(5(3e^x + 4) - 3e^x)/(3e^x + 4) dx`
= `5 intdx - int (3e^x)/(3e^x + 4] dx`
∴ `"I" = 5x - log|3e^x + 4| + c` ... [∵ `int(f'(x))/f(x)dx = log |f (x)| + c`]
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