What is the value of the Dirichlet integral?

Conditions

• The integral is an improper Riemann integral (or Lebesgue integral).
• The lower limit is 0 and the upper limit is $\infty$.
• The integrand is $\frac{\sin x}{x}$.

Reasoning, step by step

1. Identify the target integral: $\int_{0}^{\infty} \frac{\sin x}{x} dx$.
2. Recall the result derived in the video: $\frac{\pi}{2}$.
3. Note that the video presents this as the primary goal at the beginning.
4. Verify the consistency with the squared form mentioned: $\int_{0}^{\infty} \left(\frac{\sin x}{x}\right)^2 dx = \frac{\pi}{2}$.
5. Understand that this result is obtained via the Fourier transform of the rectangular pulse $f(t)$, where $F(\omega) = 2 \frac{\sin \omega}{\omega}$, and the inverse transform relation at $t=0$ leads to $\int_{-\infty}^{\infty} \frac{\sin \omega}{\omega} d\omega = \pi$, which halves to $\frac{\pi}{2}$ for the positive axis.

Example

The whiteboard explicitly states at the start: '$\int_{0}^{\infty} \frac{\sin x}{x} dx = \int_{0}^{\infty} (\frac{\sin x}{x})^2 dx = \frac{\pi}{2}$'.

Common misconceptions

• Assuming the integral diverges because $\frac{\sin x}{x}$ oscillates; it converges conditionally.
• Confusing the value $\frac{\pi}{2}$ with $\pi$; $\pi$ is the value of the integral over the entire real line $(-\infty, \infty)$.
• Thinking the squared integral has a different value; the video asserts they are equal.

Watch the explanation

Explore next

• How is the Dirichlet integral evaluated using Fourier transforms?
• What is the value of the integral over the entire real line?
• Why does the squared form have the same value?