What is the antiderivative of sin²(x)?

Solution:

The antiderivative of a function is the integral of the function.

To integrate ∫ sin² (x) dx we will use integration by part.

Also, sin² (x) = sin x × sin x

Let I = ∫ sin² (x) dx

⇒ I =  ∫ sin x × sin x dx

Using integration by part ∫ u × v = u ∫ vdx - ∫ (u' ∫ vdx ) dx

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

⇒ I = - sin x cos x + ∫ cos² x dx

⇒ I = - sin x cos x + ∫ ( 1 - sin² x ) dx

⇒ I = - sin x cos x + ∫1 - ∫ ( sin² x ) dx

⇒ I =  - sin x cos x + x - I

⇒ 2 I = - sin x cos x + x

⇒ I = x/2 - (sin x cos x) / 2

Thus, the antiderivative of sin² x is x/ 2 - (sinx cosx) / 2.

What is the antiderivative of sin²(x)?

Summary:

The antiderivative of sin² x is x/ 2 - (sinx cosx) / 2.