For the following functions find a convergent Fourier integr

For the following functions, find a convergent Fourier integral. If no such integral exists, state why.

f(x)=e^(-abs(x))cos(x) -inf<x<inf

f(x)=e^(-x)cos(x) 0<x<inf

f(x)=e^(-x)sin(x) -inf<x<inf

Solution

There is no any integral exists because of improper integral. Improper integral is very hard to study since it is very difficult to find an antiderivative of the given fuction.