What is the result of ∫ sin^2(x) dx?

To find the integral of sin²(x) with respect to x, we can use the identity involving the double angle formula for cosine. The identity states that sin²(x) can be rewritten as:

[

\sin^2(x) = \frac{1 - \cos(2x)}{2}

]

Now, substituting this back into the integral:

[

\int \sin^2(x) , dx = \int \frac{1 - \cos(2x)}{2} , dx = \frac{1}{2} \int (1 - \cos(2x)) , dx

]

This integral can be separated into two parts:

[

\int (1 - \cos(2x)) , dx = \int 1 , dx - \int \cos(2x) , dx

]

Calculating each integral separately:

1. The integral of 1 with respect to x is simply x.

2. For the integral of cos(2x), we apply the substitution u = 2x, leading to du = 2dx, or dx = (1/2)du. Thus,

[

\int \cos(