Integral of ( sin(x) )^2/x^2 from 0 to infinity 

The integral you're asking for is:

\[
I = \int_0^\infty \frac{\sin^2(x)}{x^2} \, dx
\]

To solve this, we can use a known result from definite integrals involving trigonometric functions. Specifically, there's a well-known result that:

\[
\int_0^\infty \frac{\sin^2(x)}{x^2} \, dx = \frac{\pi}{2}
\]

This result is often derived using contour integration or Fourier transform techniques. The integral converges because the sine function oscillates and the \( \frac{1}{x^2} \) factor ensures that the integral behaves properly at infinity.

Thus, the value of the integral is:

\[
\boxed{\frac{\pi}{2}}
\]

Would you like a more detailed explanation of the derivation, or do you have any other questions?

Here are 5 related questions you might be interested in:

1. What is the integral of \( \frac{\sin(x)}{x} \) from 0 to infinity?
2. How do you compute the integral \( \int_0^\infty \frac{\sin^2(ax)}{x^2} \, dx \) for any \( a > 0 \)?
3. What is the relationship between Fourier transforms and integrals of trigonometric functions?
4. How does contour integration work, and where is it used in evaluating integrals like this?
5. How can we evaluate \( \int_0^\infty \frac{\sin^2(x)}{x^3} \, dx \)?

**Tip:** Trigonometric integrals often involve symmetry and oscillatory behavior, making them candidates for evaluation using special techniques like Fourier analysis or contour integration.

Explore the integral of (sin(x))^2/x^2 from 0 to infinity with a detailed step-by-step solution. Learn how to evaluate this integral using Fourier transform techniques and contour integration. Suitable for advanced undergraduate students and anyone interested in advanced calculus problems.

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