How do you integrate cos² x by integration by parts method?

Integration by parts is a method used for integrating the functions in multiplication. To integrate ∫ cos² x, we will write cos² x = cos x × cos x.

Answer: cos² x by integration by parts method gives 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

u = cos x , v = cos x

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 dx - ∫ (cos² x) dx

I = cos x × sin x + x - I + c, ∵ I = ∫ (cos² x) dx

On solving this we get,

2I = cos x × sin x + x + c

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