Differentiating f(x) results f'(x)= (2x-4)/(3x+5). Find f(x).

To determine the function f(x), we'll have to integrate
(2x-4)/(3x+5)

Int (2x-4)dx/(3x+5) = f(x) +
C

To determine the integral of f'(x), we'll use
substitution technique, by changing the variable x.

We'll
note 3x + 5 = t

We'll subtract 5 both
sides:

3x = t-5

We'll divide
by 3:

x = (t-5)/3

We'll
differentiate both sides:

dx =
dt/3

We'll substitute in original
integral:

Int (2x-4)dx/(3x+5) = Int
[2((t-5)/3)-4]dt/3t

(2/3)*Int [(t-5)/3 - 2]dt/t = (2/3)*Int
(t-5-6)dt/3t

(2/9)*Int
(t-11)dt/t

We'll use the additive property of the
integral:

(2/9)*Int (t-11)dt/t = (2/9)*Int tdt/t -
(2/9)*Int 11dt/t

(2/9)*Int (t-11)dt/t = (2/9)*Int dt - 22/9
Int dt/t

(2/9)*Int (t-11)dt/t = 2t/9 - (22/9)*ln |t| +
C

Int (2x-4)dx/(3x+5) = 2(3x+5)/9 -
(22/9)*ln|3x+5| + C