let I=∫eaxcos(bx) dx
integrate by taking eax as a first function
I=1/b eaxsinbx -a/b∫eaxsinbx dx
now againt integrate bt taking eax as first function
I=1/b eaxsinbx +a/b2 eaxcos(bx) -a2/b2∫eaxcos(bx) dx
I=1/b eaxsinbx +a/b2 eaxcos(bx) -Ia2/b2
I= eax/(a2+b2) {a cosbx +b sin bx} +c