Integration by parts:
intudv=uv-intvdu
I=inte^(2x)sin 7xdx
e^(2x)=u => 2e^(2x)dx=du
dv=sin 7xdx => v=intsin 7xdx=-1/7cos 7x
I=-1/7e^(2x)cos 7x+2/7inte^(2x)cos 7xdx
Again:
e^(2x)=u => 2e^(2x)dx=du
dv=cos 7xdx => v=intcos 7xdx=1/7sin 7x
I=-1/7e^(2x)cos 7x+2/7[1/7e^(2x)sin7x-2/7inte^(2x)sin7xdx]
I=-1/7e^(2x)cos 7x+2/49e^(2x)sin7x-4/49inte^(2x)sin7xdx
I=-1/7e^(2x)cos 7x+2/49e^(2x)sin7x-4/49I
I+4/49I=-1/7e^(2x)cos 7x+2/49e^(2x)sin7x
53/49I=-1/7e^(2x)cos 7x+2/49e^(2x)sin7x
I=-7/53e^(2x)cos 7x+2/53e^(2x)sin7x+C
I=1/53e^(2x)(2sin 7x-7cos 7x)+C