Since #sin(2x) = 2sin x cos x#, we can write the integral in the form
#I/2 = 1/2 int e^x sin(2x) dx#
We will now carry out integration by parts
#I = e^x int sin(2x) dx-int (d/dx e^x times int sin(2x)dx)dx#
#qquad = -1/2e^x cos(2x)+1/2int e^x cos(2x) dx#
Now, we carry out integration by parts again to get
#int e^x cos(2x) dx = e^x int cos(2x) dx-int (d/dx e^x times int cos(2x)dx)dx #
#qquad = 1/2 e^x sin(2x) -1/2int e^x sin(2x)dx#
#qquad = 1/2 e^x sin(2x) -1/2I#
Thus
#I = -1/2e^x cos(2x)+1/2( 1/2 e^x sin(2x) -1/2I) implies#
#(1+1/4)I = -1/2e^x cos(2x)+ 1/4 e^x sin(2x) implies#
#I = -2/5e^x cos(2x)+1/5 e^x sin(2x)+C#
