Find the antiderivative of the function f(x)=(x+1)/(x^2+2x) .

We'll determine the indefinite integral of the given
function:

Int f(x)dx = Int
(x+1)dx/(x^2+2x)

We notice that if we'll differentiate the
denominator of the function, we'll get the numerator multiplied by
2.

We'll substitute the denominator by
t.

x^2+2x = t

We'll
differentiate both sides:

(2x + 2)dx =
dt

We'll divide by 2:

(x +
1)dx = dt/2

We'll re-write the integral in
t:

Int f(x)dx = Int dt/2t = (1/2)*ln |t| +
C

Int f(x)dx = (1/2)*ln |x^2+2x| +
C

Int f(x)dx = ln sqrt (x^2+2x) +
C