To evaluate the integral:

$\int \left( x + 1 + \frac{2}{x} - \frac{2}{x+1} \right) dx$

we can break it down into simpler integrals:

1. $\int x \, dx$
2. $\int 1 \, dx$
3. $\int \frac{2}{x} \, dx$
4. $\int \frac{-2}{x+1} \, dx$

Step-by-step solution:

1. $\int x \, dx = \frac{x^2}{2}$
2. $\int 1 \, dx = x$
3. $\int \frac{2}{x} \, dx = 2 \ln|x|$
4. $\int \frac{-2}{x+1} \, dx = -2 \ln|x+1|$

Now, putting it all together:

$\int \left( x + 1 + \frac{2}{x} - \frac{2}{x+1} \right) dx = \frac{x^2}{2} + x + 2 \ln|x| - 2 \ln|x+1| + C$

where $C$ is the constant of integration.

This is the final answer.

Do you want more details on any part of this solution?

Relative Questions:

1. How does the natural logarithm arise in integrals like $\int \frac{1}{x} dx$?
2. What happens if the integrals contain more complex rational functions?
3. Can the same method be used for trigonometric integrals?
4. How do constants affect the solution of an integral?
5. How do indefinite integrals differ from definite integrals?

Tip: When dealing with fractions in integrals, always check if you can simplify by breaking the expression into more manageable parts!