Question

If it is known that \[ \int_0^{\infty} \frac{e^{ix}}{x} dx = \frac{\pi i}{2}, \] then compute: \[ \int_0^{\infty} \frac{\sin 5x}{x} dx = ? \]

• A. \( \frac{\pi}{10} \)
• B. \( \frac{5\pi}{2} \)
• C. \( \frac{\pi}{2} \)
• D. \( \frac{\pi}{2} + 5 \)

Hint:

The integral \( \int_0^{\infty} \frac{\sin(ax)}{x} dx = \frac{\pi}{2} \) is a standard result and is independent of \( a \), for \( a>0 \).

Answer 1

The Correct Option is C

We use the result: \[ \int_0^{\infty} \frac{\sin(ax)}{x} dx = \frac{\pi}{2}, \quad \text{for } a>0 \] This is known as Dirichlet's integral and is independent of the coefficient \( a \), provided the integral converges. Hence, \[ \int_0^{\infty} \frac{\sin 5x}{x} dx = \frac{\pi}{2} \]