The integral of $\frac{\sin^2(x)}{x^2}$ is indeed a challenging one, and it can be expressed in terms of a special function known as the Sine Integral function, denoted by $\text{Si}(x)$.

The integral is:

$\int \frac{\sin^2(x)}{x^2} \, dx = \frac{\pi}{2} - \frac{\sin(2x)}{2x} - \text{Si}(2x)$

where $\text{Si}(x)$ is defined as:

$\text{Si}(x) = \int_0^x \frac{\sin(t)}{t} \, dt$

This integral cannot be expressed in terms of elementary functions alone, but the result involving the Sine Integral function provides a complete and exact form.

Would you like a step-by-step derivation or any further explanation?

Here are some related questions you might find interesting:

1. What is the integral of $\frac{\sin(x)}{x}$?
2. How do you derive the Sine Integral function $\text{Si}(x)$?
3. Can you calculate $\int \frac{\sin^2(x)}{x^2} \, dx$ numerically for specific values of $x$?
4. What are the applications of the Sine Integral function in physics?
5. How do you evaluate improper integrals involving the Sine Integral function?

Tip: When dealing with integrals that involve trigonometric functions and powers of $x$, it's often useful to check if special functions like the Sine Integral or Cosine Integral are involved.