Determine f(x) if f'(x) is (5x-3)/(x^2+4)

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

Int  (5x-3)dx/(x^2+4) = f(x) +
C

We'll use the additive property of the
integral:

Int  (5x-3)dx/(x^2+4) = Int 5xdx/(x^2+4) - Int
3dx/(x^2+4)

We'll note Int 5xdx/(x^2+4) = 5
I1

                 Int 3dx/(x^2+4) =
3I2

We'll calculate I1 using substitution
technique.

We'll note x^2+4 =
t

We'll differentiate both
sides:

2xdx = dt

We'll
substitute in the original integral and we'll get:

I1 = Int
dt/t

I1 = ln |t| + C, but t = x^2+4 >
0

I1 = ln (x^2+4) + C

5I1 =
5ln (x^2+4) + C

5I1 = ln (x^2+4)^5 +
C

We'll calculate
I2:

I2 = Int dx/(x^2+4)

I2  =
[arctan (x/2)]/2 + C

3I2 = 3[arctan (x/2)]/2
+ C

Int  (5x-3)dx/(x^2+4) = ln
(x^2+4)^5 + 3[arctan (x/2)]/2 + C