We use the integration by parts
#intuv'dx=uv-intu'vdx#
Here,
#u=sin4x#, #=>#, #u'=4cos4x#
#v'=e^(-x)#, #=>#, #v=-e^(-x)#
#inte^(-x)sin4xdx=-e^(-x)sin4x-int4*-e^(-x)cos4xdx#
#=-e^(-x)sin4x+4inte^(-x)cos4xdx#
For the integral #inte^(-x)cos4xdx#, we apply the integration by parts a second time
#u=cos4x#, #=>#, #u'=-4sin4x#
#v'=e^(-x)#, #=>#, #v=-e^(-x)#
#inte^(-x)cos4xdx=-e^(-x)cos4x-4inte^(-x)sin4xdx#
Putting it all together
#inte^(-x)sin4xdx=-e^(-x)sin4x+4(-e^(-x)cos4x-4inte^(-x)sin4xdx)#
#=-e^(-x)sin4x-4e^(-x)cos4x-16inte^(-x)sin4xdx#
Therefore,
#17inte^(-x)sin4xdx=-e^(-x)(sin4x+4cos4x)#
#inte^(-x)sin4xdx=(-e^(-x)(sin4x+4cos4x))/17+C#