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^2 + 1/x] / (x+1)

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

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

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

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

=> x - 1 + 1/x

**Therefore x/ (1+1/x) + (1/x)/ (x + 1) = x - 1 + 1/x**

To simplify x/ (1+1/x) + (1/x)/ (x + 1).

x/(x+1/x) = x*x/x((x+1/x) = x^2/(x+1)...(1)

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

The two denominators (x+1) and x(x+1) have the LCM x(x+1).

So we write equivalent rational form of x^2/(x+1) at (1) with denomiator x(x+1) as below:

So x^2/(x+1) = x^3/x(x+1)...(3).

Therefore,

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

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

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

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

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**