I´ll start by integrating e^(2x)e2x and leaving cos(x)cos(x) as it is and then derive it leaving the integrated part as it is.
e^(2x)/2*cos(x)-inte^(2x)/2*(-sin(x))dx=e2x2⋅cos(x)−∫e2x2⋅(−sin(x))dx=
=e^(2x)/2*cos(x)+1/2inte^(2x)*(sin(x))dx==e2x2⋅cos(x)+12∫e2x⋅(sin(x))dx=
by parts again:
=e^(2x)/2*cos(x)+1/2[e^(2x)/2*(sin(x))-inte^(2x)/2*cos(x)dx]==e2x2⋅cos(x)+12[e2x2⋅(sin(x))−∫e2x2⋅cos(x)dx]=
=e^(2x)/2*cos(x)+e^(2x)/4*(sin(x))-1/4inte^(2x)*cos(x)dx=e2x2⋅cos(x)+e2x4⋅(sin(x))−14∫e2x⋅cos(x)dx
So your integral is:
inte^(2x)cos(x)dx=e^(2x)/2*cos(x)+e^(2x)/4*(sin(x))-1/4inte^(2x)*cos(x)dx∫e2xcos(x)dx=e2x2⋅cos(x)+e2x4⋅(sin(x))−14∫e2x⋅cos(x)dx
Now I can take to the left the integral: -1/4inte^(2x)*cos(x)dx−14∫e2x⋅cos(x)dx
Giving:
inte^(2x)cos(x)dx+1/4inte^(2x)*cos(x)dx=e^(2x)/2*cos(x)+e^(2x)/4*(sin(x))∫e2xcos(x)dx+14∫e2x⋅cos(x)dx=e2x2⋅cos(x)+e2x4⋅(sin(x))
5/4inte^(2x)*cos(x)dx=e^(2x)/2*cos(x)+e^(2x)/4*(sin(x))54∫e2x⋅cos(x)dx=e2x2⋅cos(x)+e2x4⋅(sin(x))
and finally:
inte^(2x)*cos(x)dx=4/5[e^(2x)/2*cos(x)+e^(2x)/4*(sin(x))]+c∫e2x⋅cos(x)dx=45[e2x2⋅cos(x)+e2x4⋅(sin(x))]+c
