Fourier Transform - not sure if I totally understand?

So the question states:

Find the fourier transform of the function f defined by:

f(x) = -e^(ax) x<0

= e^(-ax) x>0

I think I know what I'm doing, but this is totally new to me, and confirmation that I know what I'm doing would be awesome!!!!

So my answer is:

the transform of e^(-ax) is:

f hat k = integral(0 to infinity) e^(-ax) e^(-ikx) dx

= integral(0 to infinity) e^-(a+ik)x dx

= 1/(a+ik)

the transform of -e^(ax) is:

f hat k = - integral(0 to infinity) e^(ax) e^(-ikx) dx

= - integral(0 to infinity) e^(a-ik)x dx

= - 1/(a-ik)

so total answer is 1/(a+ik) - 1/(a-ik)

So cross multiplying with congugates, gives:

2ik/(a^2+k^2)

Is this correct?

Any help would be hugely appreciated! Thanks!