Integrate by parts:
#int sinx e^(-x) dx = int sinx d/dx(-e^(-x)) dx#
#int sinx e^(-x) dx = -e^(-x)sinx + int e^(-x) d/dx(sinx ) dx#
#int sinx e^(-x) dx = -e^(-x)sinx + int e^(-x) cosx dx#
and then again:
#int sinx e^(-x) dx = -e^(-x)sinx + int cosx d/dx (-e^(-x))dx#
#int sinx e^(-x) dx = -e^(-x)sinx -e^(-x)cosx + int e^(-x) d/dx(cosx )dx#
#int sinx e^(-x) dx = -e^(-x)sinx -e^(-x)cosx - int sinx e^(-x) dx#
The same integral appears on both sides and we can solve fro it:
#2int sinx e^(-x) dx = -e^(-x)sinx -e^(-x)cosx + C#
#int sinx e^(-x) dx = -(e^(-x)(sinx +cosx ))/2+ C#
