To integrate the function sin²(x), we use a trigonometric identity to simplify the process. The identity we use is:
sin²(x) = (1 – cos(2x)) / 2
Using this identity, we can rewrite the integral:
∫ sin²(x) dx = ∫ (1 – cos(2x)) / 2 dx
This can be separated into two simpler integrals:
∫ sin²(x) dx = 1/2 ∫ dx – 1/2 ∫ cos(2x) dx
Now, we integrate each part:
1. The integral of dx is simply x.
2. The integral of cos(2x) dx is (1/2)sin(2x), since the derivative of sin(2x) is 2cos(2x).
Putting it all together, we have:
∫ sin²(x) dx = 1/2 x – 1/4 sin(2x) + C
where C is the constant of integration. Therefore, the final result is:
∫ sin²(x) dx = (1/2)x – (1/4)sin(2x) + C