What is the method for evaluating the Fourier transform of a given ODE?

[sachi]

we need to find the F.T of
f(t) = 0 for t<0
f(t) = exp(-at) for t>=0
where a is a real positive constant
and F(w) = the integral w.r.t t between minus infinity and plus infinity of [exp(iwt)*f(t)]

which turns out to be 1/(a-iw)

we now have the ODE L*dI/dt + RI = f(t)
where L,R are constants represeting resistance and inductance. we need to show that the Fourier transform of I(t) is 1/(a-iw)(R-Liw) which is again straightforward. We need to find I(t). I have sepearated the F.T of I(t) using partial fractions and used the inverse Fourier transform, but I'm not sure how to evaluate the integrals.
thanks very much for your help

 

[vela]

To evaluate an integral like
$$\int_{-\infty}^\infty \frac{e^{-i\omega t}}{a-i\omega}\,d\omega,$$ you typically use contour integration in the complex plane and the residue theorem. I'm guessing you were expected, however, to invert the Fourier transforms by looking them up in a table of Fourier transforms.