Integration by parts states that:
#intudv=uv-intvdu#
We let #u=e^-x# and #dv=cos(3x)#
We integrate #u# and integrate #dv#
#=>du=e^-x*d/dx[-x]#
#=>du=-e^-x#
#=>v=intcos(3x)dx#
#=>v=sin(3x)/3#
Substitute:
#e^-x*sin(3x)/3-intsin(3x)/3*-e^-xdx#
#=>e^-x*sin(3x)/3+1/3intsin(3x)*e^-xdx#
Repeat the process.
#u=e^-x#
#dv=sin(3x)#
#=>du=-e^-x#
#=>v=intsin(3x)dx#
#=>v=-cos(3x)/3#
#=>e^-x*sin(3x)/3+1/3[e^-x*-cos(3x)/3-int-cos(3x)/3*-e^-xdx]#
#=>e^-x*sin(3x)/3+1/3[e^-x*-cos(3x)/3-1/3intcos(3x)*e^-xdx]#
#=>e^-x*sin(3x)/3+e^-x*-cos(3x)/9-1/9intcos(3x)*e^-xdx#
That integral is our original integral!
We let #s=intcos(3x)*e^-xdx#
#=>s=e^-x*sin(3x)/3+e^-x*-cos(3x)/9-1/9s#
#=>9s=3e^-x*sin(3x)+e^-x*-cos(3x)-s#
#=>10s=3e^-x*sin(3x)+e^-x*-cos(3x)#
#=>s=(3e^-x*sin(3x)+e^-x*-cos(3x))/10#
#=>s=(3e^-x*sin(3x)-e^-xcos(3x))/10#
#=>s=(e^-x(3sin(3x)-cos(3x)))/10# Do you #C# why this is incomplete?
#=>inte^-xcos(3x)dx=(e^-x(3sin(3x)-cos(3x)))/10+C#
