(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let x(t) = e^{-100t}u(t)

u(t) = 0 for t < 0

u(t) = 1 for t > 0

Evaluate the following integral (from -∞ to ∞):

X(ω) = ∫ x(t)e^{-iωt}dt

2. Relevant equations

See below.

3. The attempt at a solution

I tried to evaluate the integral by splitting it in two parts, since x(t) takes two different values. Keep in mind I'm replacing ∞ with a to evaluate the limit later.

from -∞ to 0:

X1(ω) = ∫ e^{-iωt}dt

X1(ω) = (1/-iω) e^{-iωt}(from -∞ to 0)

X1(ω) = (1/-iω)(1 - e^{iωa})

from 0 to ∞:

X2(ω) = e^{-100}∫ e^{-iωt}dt

X2(ω) = e^{-100}(1/-iω) e^{-iωt}(from 0 to ∞)

X2(ω) = e^{-100}(1/-iω)(e^{-iωa}- 1)

X(ω) = X1(ω) + X2(ω) = (1/-iω)(1 - e^{iωa}) + e^{-100}(1/-iω)(e^{-iωa}- 1)

= (1/-iω) (1 - e^{iωa}+ e^{-100}e^{-iωa}- e^{-100})

This is where I'm stuck. The answer is supposed to be X(ω) = 1 / (100 + iω), and I have absolutely no idea how I'm supposed to get there. Would someone mind helping me out?

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# Please help me evaluate this seemingly simple integral

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