Integrate $\left(x - \frac{5}{2x}\right)^2$ with respect to $x$.

1. Expand the integrand

$\left(x - \frac{5}{2x}\right)^2 = x^2 - 2(x)\left(\frac{5}{2x}\right) + \left(\frac{5}{2x}\right)^2$
$= x^2 - 5 + \frac{25}{4x^2}$

2. Rewrite the expression with negative indices for integration

$= x^2 - 5 + \frac{25}{4}x^{-2}$

3. Integrate term by term

$\int \left(x^2 - 5 + \frac{25}{4}x^{-2}\right) dx$
$= \frac{x^3}{3} - 5x + \frac{25}{4}\left(\frac{x^{-1}}{-1}\right) + c$
$= \frac{x^3}{3} - 5x - \frac{25}{4x} + c$