The antiderivative of sin² x can be found using a common trigonometric identity. First, we can utilize the identity:
sin² x = (1 – cos(2x)) / 2
Thus, we can rewrite the integral of sin² x as:
∫sin² x dx = ∫(1 – cos(2x)) / 2 dx
Next, we can separate the integral:
∫sin² x dx = 1/2 ∫(1 – cos(2x)) dx
This simplifies to:
= 1/2 (∫1 dx – ∫cos(2x) dx)
Now, evaluating each part:
∫1 dx = x and ∫cos(2x) dx = (1/2)sin(2x) (using a simple substitution technique).
Putting everything together, we have:
∫sin² x dx = 1/2 (x – (1/2)sin(2x)) + C
So, the final result can be expressed as:
∫sin² x dx = (x/2) – (1/4)sin(2x) + C
where C is the constant of integration.