If f'(x) = ( 2x+3) /x find f(x) if f(1) = 3

Given the first
derivative:

f'(x) =
(2x+3)/x

We need to find
f(x).

First we will simplify
f'(x).

==> f'(x) = 2x/x  +
3/x

==> f'(x) = 2 +
3/x

Now we know that f(x) = intg
f'(x).

==> f(x) = intg ( 2 + 3/x)
dx

             = intg 2 + intg 3/x 
dx

              = 2x + 3*lnx +
C

==> f(x) = 2x + 3*lnx +
C

But we are given that f(1) =
3

==> 2 + 3ln1 + C =
3

==> But ln 1 =
0

==> 2 + C =
3

==> C = 1

Then we
conclude that:

f(x) = 2x + 3*lnx + 1