We have to simplify: x/ (1+1/x) + (1/x)/ (x + 1)

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

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

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

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

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

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

**The simplified form of x/ (1+1/x) + (1/x)/ (x + 1) is (x^2 - x + 1)/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)

(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 applying the formula:

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