First, expand the integrand:

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

Now, integrate each term separately:

\(\int x^2 \, dx = \frac{x^3}{3} + C_1\)

\(\int 2 \, dx = 2x + C_2\)

\(\int \frac{1}{x^2} \, dx = \int x^{-2} \, dx = -x^{-1} + C_3 = -\frac{1}{x} + C_3\)

Combine the results:

\(\int \left( x + \frac{1}{x} \right)^2 \, dx = \frac{x^3}{3} - \frac{1}{x} + 2x + C\)