Question

The value of \( \int_0^\infty \frac{\sin(4x)}{\pi x} \, dx \) is ________

Hint:

The integral \( \int_0^\infty \frac{\sin(ax)}{\pi x} \, dx \) is a standard result from Fourier analysis and is commonly used in signal processing. For any positive constant \( a \), the integral evaluates to \( \frac{1}{2} \).

Answer 1

We are asked to evaluate the following integral: \[ I = \int_0^\infty \frac{\sin(4x)}{\pi x} \, dx \] Step 1: Recognize the standard integral form. 
This integral is a standard form known as the sine integral, often found in the context of Fourier transforms. Specifically, the general form is: \[ \int_0^\infty \frac{\sin(ax)}{\pi x} \, dx = \frac{1}{2} \quad \text{for any constant } a > 0. \] Step 2: Apply the standard result. 
In our case, \( a = 4 \). So applying the result, we get: \[ \int_0^\infty \frac{\sin(4x)}{\pi x} \, dx = \frac{1}{2}. \] Thus, the value of the integral is \( 0.50 \). 
Step 3: Conclusion. 
The value of the integral is approximately 0.50.