I am working through the above question in preparation for an upcoming exam. I have completed part (a) and quoted the inversion formula for part (b), but I cannot see how to find a form to evaluate the integral. Can anyone point me in the right direction?
Answer for part (a) should be $$\frac{1-ik}{1+k^2}$$
So by the inversion formula $$f(x)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\frac{1-ik}{1+k^2} e^{ikx}\,dk$$
Now separate the integrand into real part and imaginary part. Notice that $f(x)$ is a real function, so the imaginary part in the integrand should be zero. The real part is exactly the integral in part (b). Can you go on from here?