Am I doing this fourier transform right?

Let's say I have a function$$f1(t)=e ^{-5t}u(t)$$

Based from the transform pair table its fourier transform should be $$\frac{ 1 }{ 5+jw }$$

What if I solved for the fourier transform of
$$f1(t)e^{j5t}$$
The properties of fourier transform say that this is called "multiplied by a complex exponential" and its transform is

$$X(w-w_{0})$$
So what i did is i got the fourier transform of f1 and just changed the value of w to w-w0 where w0 is equal to 5

It would look like
\[\frac{ 1 }{ 5+j(w-5) }]\
Is this correct? or am I missing something?

Question

Am I doing this fourier transform right?

Let's say I have a function$$f1(t)=e ^{-5t}u(t)$$

Based from the transform pair table its fourier transform should be $$\frac{ 1 }{ 5+jw }$$

What if I solved for the fourier transform of
$$f1(t)e^{j5t}$$
The properties of fourier transform say that this is called "multiplied by a complex exponential" and its transform is

$$X(w-w_{0})$$
So what i did is i got the fourier transform of f1 and just changed the value of w to w-w0 where w0 is equal to 5

It would look like 
\[\frac{ 1 }{ 5+j(w-5) }]\
Is this correct? or am I missing something?