#I=int\ e^x cos(x)\ "d"x#
We will be using integration by parts, which states that #int\ u\ "d"v=uv-int\ v\ "d"u#.
Use integration by parts, with #u=e^x#, #du=e^x\ "d"x#, #"d"v=cos(x)\ "d"x#, and #v=sin(x)#:
#I=e^xsin(x)-int\ e^xsin(x)\ "d"x#
Use integration by parts again to the second integral, with #u=e^x#, #"d"u=e^x\ "d"x#, #"d"v=sin(x)\ "d"x#, and #v=-cos(x)#:
#I=e^xsin(x)+e^xcos(x)-int\ e^xcos(x)\ "d"x#
Now, recall we defined #I=int\ e^x cos(x)\ "d"x#. Thus, the above equation becomes the following (remembering to add a constant of integration):
#I=e^xsin(x)+e^xcos(x)-I+C#
#2I=e^xsin(x)+e^xcos(x)+C=e^x(sin(x)+cos(x))+C#
#I=1/2e^x(sin(x)+cos(x))+C#