To solve the integral $\int \left( \frac{1}{x} + 1 \right)^2 dx$, let’s proceed step by step:

Step 1: Expand the square

$\left( \frac{1}{x} + 1 \right)^2 = \frac{1}{x^2} + 2\frac{1}{x} + 1$

So the integral becomes: $\int \left( \frac{1}{x^2} + 2\frac{1}{x} + 1 \right) dx$

Step 2: Separate the integral

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

Step 3: Compute each term

1. First term: $\int \frac{1}{x^2} dx$ Recall that $\frac{1}{x^2} = x^{-2}$, so: $\int x^{-2} dx = \frac{x^{-1}}{-1} = -\frac{1}{x}$

2. Second term: $2 \int \frac{1}{x} dx$ The integral of $\frac{1}{x}$ is $\ln|x|$, so: $2 \int \frac{1}{x} dx = 2 \ln|x|$

3. Third term: $\int 1 dx$ The integral of $1$ is $x$, so: $\int 1 dx = x$

Step 4: Combine the results

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

Where $C$ is the constant of integration.

Would you like me to expand on any part of this solution or verify using an alternate method? Here are some related questions to explore:

1. What is the physical interpretation of logarithmic terms in integrals?
2. How does the method of substitution work for similar rational integrals?
3. What happens if we apply integration by parts to this problem?
4. Could we solve this using definite integral bounds instead of indefinite?
5. How do powers of $\frac{1}{x}$ behave as $x \to 0$ or $x \to \infty$?

Tip: Always simplify your integrand when possible before integrating, as it often makes the process smoother!