What is the Fourier Transform of $\DeclareMathOperator{\rect}{rect} \rect(2Bt)\cos[{\omega}_Ct + k_fm(t_k)t] $?
If you define the $\textrm{rect}()$ function according to this definition, then your result is correct.
However, there might be a problem with the definition of the $\textrm{rect}()$ function. If in your book they define that function to have unit area, then the result of the book would be correct.