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## Homework Statement

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

## Homework Equations

See below.

## 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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