How is sin^2(x) integrated?

The integral of sin^2(x) is (1/2)x - (1/4)sin(2x) + C.

To integrate sin^2(x), we can use the identity sin^2(x) = (1/2)(1-cos(2x)). Therefore, the integral of sin^2(x) can be written as:

∫sin^2(x) dx = ∫(1/2)(1-cos(2x)) dx

We can then split this integral into two parts:

∫(1/2) dx - ∫(1/2)cos(2x) dx

The first integral is simply (1/2)x + C. For the second integral, we can use the substitution u = 2x, du/dx = 2, and dx = (1/2)du. This gives us:

∫(1/2)cos(2x) dx = (1/4)∫cos(u) du

Using the identity ∫cos(u) du = sin(u) + C, we get:

∫(1/2)cos(2x) dx = (1/4)sin(2x) + C

Putting these two parts together, we get the final answer:

∫sin^2(x) dx = (1/2)x - (1/4)sin(2x) + C