Simplify x/ (1+1/x) + (1/x)/ (x + 1).

We'll take the denominator of the first ratio and we'll multiply
1 by x:

1+1/x = (x+1)/x

We'll multiply
the numerator x of the first ratio by the inversed
denominator:

x*x/(x+1) =
x^2/(x+1) (1)

We'll re-write the second
ratio:

(1/x)/ (x + 1) = 1/x(x+1)
(2)

We'll add (1) + (2)

 x^2/(x+1) +
1/x(x+1)

We'll multiply by x the first
ratio:

(x^3 + 1)/x(x+1)

We'll re-write
the sum of cubes:

x^3 + 1 = (x+1)(x^2 - x +
1)

(x^3 + 1)/x(x+1) = (x+1)(x^2 - x +
1)/x(x+1)

We'll simplify:

(x+1)(x^2 - x
+ 1)/x(x+1) = (x^2 - x + 1)/x

(x^2 - x + 1)/x = x^2/x - x/x +
1/x

(x^2 - x + 1)/x = x - 1 +
1/x