what are and why are sine and cosine modulated integrals used?
As someone made me notice in the comments, the integrals are just the coefficients of the Fourier Series for that frequency component (when the series is expressed with the cosine and sine parts separately).

The integral is performed on a period much bigger then the period corresponding to the frequency of the disturbance so that a better averaged is obtained.

To get back the signal in the time domain it is sufficient to multiply the two components by the cosine and sine respectively. Of course in this way the signal has only one frequency component.