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Substituting $t = 0$ into the inverse Fourier transform formula simplifies the exponential term $e^{i\omega t}$ to 1, removing the oscillatory component from the integrand. This transforms the inverse transform equation into a direct relationship between the value of the original function at the origin, $f(0)$, and the integral of its Fourier transform $F(\omega)$ over the entire real line.

Conditions: The inverse Fourier transform formula is $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$.; The function $f(t)$ is the rectangular pulse with $f(0)=1$.; $F(\omega)$ is the previously computed Fourier transform $2 \frac{\sin \omega}{\omega}$.

Substituting $t = 0$ into the inverse Fourier transform formula simplifies the exponential term $e^{i\omega t}$ to 1, removing the oscillatory component from the integrand. This transforms the inverse transform equation into a direct relationship between the value of the original function at the origin, $f(0)$, and the integral of its Fourier transform $F(\omega)$ over the entire real line.

Conditions: The inverse Fourier transform formula is $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$.; The function $f(t)$ is the rectangular pulse with $f(0)=1$.; $F(\omega)$ is the previously computed Fourier transform $2 \frac{\sin \omega}{\omega}$.

The value of the Dirichlet integral $\int_{0}^{\infty} \frac{\sin x}{x} dx$ is $\frac{\pi}{2}$. The video also states that the squared form $\int_{0}^{\infty} \left(\frac{\sin x}{x}\right)^2 dx$ yields the same result, $\frac{\pi}{2}$.

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 $\frac{\sin x}{x}$.

The value of the Dirichlet integral $\int_{0}^{\infty} \frac{\sin x}{x} dx$ is $\frac{\pi}{2}$. The video also states that the squared form $\int_{0}^{\infty} \left(\frac{\sin x}{x}\right)^2 dx$ yields the same result, $\frac{\pi}{2}$.

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 $\frac{\sin x}{x}$.

The video proposes two distinct mathematical methods for evaluating the Dirichlet integral: 1. The Fourier Transform method, which involves transforming a rectangular pulse function and using the inverse transform. 2.

Conditions: The problem is the evaluation of $\int_{0}^{\infty} \frac{\sin x}{x} dx$.; The methods are listed as alternatives.

The video proposes two distinct mathematical methods for evaluating the Dirichlet integral: 1. The Fourier Transform method, which involves transforming a rectangular pulse function and using the inverse transform. 2.

Conditions: The problem is the evaluation of $\int_{0}^{\infty} \frac{\sin x}{x} dx$.; The methods are listed as alternatives.