Integrate the function f(x) =cos² x.

To integrate ∫ cos² x dx, we will write cos² x = cos x × cos x and integrate using part integration.

Answer: 1 / 2 ( cos x sin x ) + x / 2 + C

Let's integrate ∫ cos² x dx.

Explanation: 

To integrate ∫ cos² x dx assume I = ∫ cos² x dx.

I = ∫ cos² x dx

I = ∫ cos x × cos x dx

We know that ∫ (u × v) dx = u ∫ v dx - ∫ ( u' - ∫ v dx) dx

I = cos x × sin x - ∫ (( - sin x ) × sin x) dx

I = cos x × sin x + ∫ ( sin² x) dx

I = cos x × sin x + ∫ ( 1 - cos² x) dx

I = cos x × sin x + ∫ 1 - ∫ (cos² x) dx

I = cos x × sin x + x - I

On solving this we get,

2I = cos x × sin x + x 

I = 1 / 2 ( cos x × sin x ) + x / 2 + C

Thus, the integral of  ∫ cos² x is 1 / 2 ( cos x sin x ) + x / 2 + C