Mathematically solving fourier transform

[Robismyname]

Since I lack the understand of real world applications of Fourier Transform in the real world I decided to buy a signals and systems book (Lathi) do some Fourier Transform problems and them do the same problem in Matlab.

The question in the book wants me to find the Fourier Transform of signal f(t) = e^-at; from 0 to T

I know in order to find FT I have to do the following:
T
step 1: F(w) = ∫ f(t) * e^-jwt dt
0

T
step 2: ∫ e^-(a+jw)t dt [combine like terms]
0


step 3: [ e^-(a+jw)0 ] - [ e^-(a+jw)T ] [integrate over 0 to T]


step 4: 1 - e^-(a+jw)T [solve]



The book says the answer is:

1-e^-(jw+a)T
----------------------------
jw+a

How did the book get the denominator section of jw+a? I can't get from step 4 to the book answer. What am I missing here?

 

[tindel]

Note that:

∫e^(-t(x+jy))*dt ≠ e^(-t(x+jy)) + C

Use u-substitution to come to the correct answer

This helped jog my memory:
http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/usubdirectory/USubstitution.html