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 FT problems and them do the same in problem in Matlab.

The question in the book wants me to find the FT 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




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 cant get from step 4 to the book answer. What am I missing here?

 

[jeffrey samuel]

you forgot something when you integrate e^ax the ans is not e^ax as you have used

it is (e^ax)/a

and in your case a=(jw+a)