Fourier Transform of a signal using direct integration and properties Am trying to compute the Fourier Transform of a function using the properties of the Fourier Transform once and checking my answer using direct integration. My problem is that am not getting the same for both ways.



Sorry for not being able to type the equations as I am not familiar on how to do that, It would be great if somebody can tell me how. Your help is appreciated. Thanks.
Your first solution using the properties of the Fourier transform is correct. Your second solution is wrong, because you forgot to include the unit step function. Your function $g(t)$ should be defined by

$$g(t)=e^{-t}u(t)\tag{1}$$

which gives for $g(2t-1)$

$$g(2t-1)=e^{-(2t-1)}u(2t-1)=e^{-(2t-1)}u\left(t-\frac12\right)\tag{2}$$

Consequently, the Fourier integral is given by

$$\mathcal{F}\{g(2t-1)\}=\int_{\frac12}^{\infty}e^{-(2t-1)}e^{-j\omega t}dt\tag{3}$$

I'm sure you can prove now that the result of $(3)$ equals the result you got from using the properties of the Fourier transform.