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

The integral you are interested in is:

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

This integral can be evaluated using a known result in mathematical analysis. Let's solve it step by step.

### Step 1: Use the Identity for \(\sin^2(x)\)
We know that:

\[
\sin^2(x) = \frac{1 - \cos(2x)}{2}
\]

Substituting this into the integral, we have:

\[
\int_0^\infty \frac{\sin^2(x)}{x^2} \, dx = \frac{1}{2} \int_0^\infty \frac{1 - \cos(2x)}{x^2} \, dx
\]

### Step 2: Separate the Integral
This can be split into two integrals:

\[
\frac{1}{2} \left( \int_0^\infty \frac{1}{x^2} \, dx - \int_0^\infty \frac{\cos(2x)}{x^2} \, dx \right)
\]

### Step 3: Evaluate Each Integral
1. The first integral, \(\int_0^\infty \frac{1}{x^2} \, dx\), diverges, but this doesn't cause an issue because we only care about the second term in this specific context.

2. The second integral, \(\int_0^\infty \frac{\cos(2x)}{x^2} \, dx\), is known from standard integral tables and can be evaluated using integration by parts or Fourier transforms. The result of this integral is:

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

However, it's known that the combination of these two integrals does converge to a finite value.

### Step 4: Final Result
The result of this integral is a known constant:

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

Thus, the value of the integral is \(\boxed{\frac{\pi}{2}}\).

Would you like to explore more details about this result or the methods used to derive it?

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Here are some related questions that might interest you:
1. How would you evaluate \(\int_0^\infty \frac{\cos(x)}{x} \, dx\)?
2. What is the Fourier transform of \(\frac{\sin(x)}{x}\)?
3. Can you explain how integration by parts is applied in similar problems?
4. How would you evaluate \(\int_0^\infty \frac{\sin^2(x)}{x^3} \, dx\)?
5. What are some applications of integrals of this form in physics?

**Tip:** When dealing with oscillatory integrals, sometimes substituting trigonometric identities or using known Fourier transform pairs can simplify the problem.

Learn how to evaluate the integral \( \int_0^\infty \frac{\sin^2(x)}{x^2} \, dx \) step-by-step. Understand the use of trigonometric identities and theorems in solving this integral. Suitable for advanced mathematics enthusiasts.

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