$\int \sin ^{2} x d x$

Step 1 ：Given the integral of \(\sin^2 x\)

Step 2 ：We can solve this using the power-reduction identity. The power-reduction identity for \(\sin^2 x\) is \(\sin^2 x = \frac{1 - \cos(2x)}{2}\)

Step 3 ：Substitute this into the integral and then solve

Step 4 ：\(x = x\)

Step 5 ：Integral = \(\frac{x}{2} - \frac{\sin(2x)}{4}\)

Step 6 ：The integral of \(\sin^2 x\) is \(\frac{x}{2} - \frac{\sin(2x)}{4}\)

Step 7 ：Final Answer: \(\boxed{\frac{x}{2} - \frac{\sin(2x)}{4}}\)