@electricalmath: Let’s integrate sin(x) / x from 0 to infinity, known as the Dirichlet integral. The integrand, sin(x) / x, is the unnormalized sinc function sinc(x) that is crucial in electrical engineering, particularly in signal processing. In the frequency domain, an ideal “brick-wall” low-pass filter is a rectangular pulse. Taking the inverse Fourier transform of this rectangular pulse gives you a sinc function in the time domain. If you integrate the sinc function up to a variable t (which gives the Sine Integral, Si(t)), you get the step response of that ideal low-pass filter. The fact that the integral overshoots its final value before settling is the exact mathematical origin of the Gibbs phenomenon — that famous ringing artifact you see when approximating square waves with Fourier series. The sinc function does not have an elementary antiderivative. To solve the integral, we introduce a decay factor e^–(ax) to create a family of integrals I(a). Then, inspired by Feynman’s technique, we differentiate under the integral sign to cancel the problematic denominator. At this point, integrating by parts becomes possible. The final step is to integrate back to get I(a), and take the limit as a approaches 0 to evaluate our target integral. #math #calculus #engineering #integral #integration

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Sunday 08 March 2026 19:27:29 GMT