What is the integral of the function y=cos^2x-sin^2x?

We'll have to use the double angle
identities:

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

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

We'll subtract (2) from
(1):

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

We'll eliminate like
terms:

(cos x)^2 - (sin x)^2 =
cos(x/2)

We'll integrate both
sides:

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

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

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

The requested integral of the difference
(cos x)^2 - (sin x)^2 is:  Int [(cos x)^2 - (sin x)^2] dx = 2sin(x/2) +
C.