Hey somebody can help me please to understand how they arrive to this solve of correlation rks,rkc. It is from a fsk demodulation algorithm they do 2 Integration, one time between sin(2*pi*fc*t+2pi*m*deltaf*t)*cos(2*pi*fc*t+2pi*m*deltaf*t+phase) and seconed time between cos(2*pi*fc*t+2pi*m*deltaf*t)*cos(2*pi*fc*t+2pi*m*deltaf*t+phase) but I doesnt understand how they solve this integral (7.5.11)

I don't think I can work the problem but maybe can give you some insight from a system point of view. In practice the modulation is done with a phase accumulator. The accumulator rotates at the carrier (or IF) frequency + the modulating signal. The modulating signal is shaped (Gaussian, Raised Cosine, etc.) to give the proper signal excursions but still have band limiting. The change of phase is steered thus modulating the frequency. The signal then is just a sine/cosine look up of the phase.

There are various ways to demod the signal. One direct way (very good) is to take the arc-tangent of the signal (table look up or CORDIC) to get back the Phase. Then take the derivative of the Phase to get the instantaneous Frequency (d Phase/ dt).

FWIW