Find the antiderivative of the function y=e^x*cos^2x .

The antiderivative of a function is the indefinite
integral of the given function.

Int f(x)dx = Int e^x*cos^2x
dx

We'll substitute the square of cosine of function, by
the formula:

(cos x)^2 = (1 + cos
2x)/2

We'll re-write the
integral:

Int e^x*cos^2x dx = Int e^x*(1 + cos
2x)dx/2

We'll apply the additive property of indefinite
integrals:

Int e^x*(1 + cos 2x)dx/2 = Int e^xdx/2 + Int
(e^x*cos 2x)dx/2

We'll note Int e^xdx/2 =
I1

Int (e^x*cos 2x)dx/2 =
I2

I1 = e^x/2 + C (1)

We'll
solve I2 by parts:

u = cos 2x => du = -2 sin
2x

dv = e^xdx => v =
e^x

I2 = u*v - Int vdu

I2 =
e^x*cos 2x + 2Int e^x*sin 2x dx

We'll solve 2Int e^x*sin 2x
dx by parts:

u = sin 2x => du = 2 cos
2x

dv = e^xdx => v =
e^x

2Int e^x*sin 2x = 2e^x*sin 2x - 4 Int e^x*cos
2xdx

But Int e^x*cos 2xdx =
I2

I2 = e^x*cos 2x + 2e^x*sin 2x - 4
I2

We'll add 4I2 both
sides:

5I2 = e^x*cos 2x + 2e^x*sin
2x

I2 = (e^x*cos 2x + 2e^x*sin 2x)/5 +
C (2)

Int e^x*(1 + cos 2x)dx/2 = (1) +
(2)

Int e^x*(1 + cos 2x)dx/2 = e^x/2 +
(e^x*cos 2x + 2e^x*sin 2x)/10 + C