I broke the cosine function into two parts:

cos^2 x * cos x

The integral now looks like this:

cos^2 x * sin x * cos x dx

I then let cos^2 x = 1 - sin^2 x.

Here is the integral now:

(1 - sin^2 x) * sin x * cos x dx

I let u = sin x making du = cos x dx.

The Integral became very easy: (1 - u^2)u du.

After integrating, I back-substituted for u.

My final answer is (1/4)(2 sin^2 x - sin^4 x) + C.

The textbook answer is (-1/4)cos^4 x + C.

Who is right?

What did I do wrong?

Is my answer equivalent to the textbook answer?