Integral of ( sin(x) )^2/x^2. Answer may involve special functions.

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.

Explore the integral of (sin(x))^2 / x^2 with a detailed solution involving the Sine Integral function. This problem delves into advanced topics in integral calculus and special functions.

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