What is the antiderivative of y=cos^3x?

To detemrine the antiderivative, we'll have to calculate
the indefinite integral of the given function.

We'll write
the integrand (cos x)^3 = (cos x)^2*cos x.

Int (cos
x)^2*cos x dx = Int [1-(sin x)^2]*cos x dx

We'll replace
sin x by t:

sin x = t

cos x dx
= dt

Int [1-(sin x)^2]*cos x dx = Int (1 -
t^2)dt

Int (1 - t^2)dt = Int dt - Int
t^2dt

Int (1 - t^2)dt= t - t^3/3 +
C

Int (cos x)^2*cos x dx = sin x - (sin x)^3/3 +
C

The requested antiderivative is: F(x) = sin
x - (sin x)^3/3 + C.