The value of \( \int_0^\infty \frac{\sin(4x)}{\pi x} \, dx \) is ________
Show Hint
Correct Answer: 0.5
Solution and Explanation
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: \[ \int_0^\infty \frac{\sin(ax)}{\pi x} \, dx = \frac{1}{2} \quad {for any constant } a>0. \] 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.
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