When integrate sin x gives -cosx. When integrate sin^2x gives cos^2x.

We'll have to use the
formula:

(sin x)^2 = [1 -
cos(x/2)]/2

We'll integrate both
sides:

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

We'll use the additive property of
integral:

Int [1 - cos(x/2)]dx/2 = Int dx/2 - Int
cos(x/2)dx/2

Int dx/2 = (1/2)/Int
xdx

Int dx/2 = (x^2)/4 + C
(1)

Int cos(x/2)dx/2 = (1/2)*Int
cos(x/2)dx

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

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

Int (sin x)^2 dx =  (1) +
(2)

Int (sin x)^2 dx = (x^2)/4 + sin(x/2) +
C

So, when integrate (sinx)^2,
the result is not (cosx)^2.