Integral of sinx^2dx dx

1. Rewrite the integrand:

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

2. Integrate term-by-term:

   1. The integral of a constant is the constant times the variable of integration:

      $$\int \frac{1}{2}\, dx = \frac{x}{2}$$

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}$$

      1. Let $u = 2 x$.

         Then let $du = 2 dx$ and substitute $\frac{du}{2}$:

         $$\int \frac{\cos{\left(u \right)}}{4}\, du$$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$$

            1. The integral of cosine is sine:

               $$\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$$

            So, the result is: $\frac{\sin{\left(u \right)}}{2}$

         Now substitute $u$ back in:

         $$\frac{\sin{\left(2 x \right)}}{2}$$

      So, the result is: $- \frac{\sin{\left(2 x \right)}}{4}$

   The result is: $\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$

3. Add the constant of integration:

   $$\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}$$

The answer is:

$$\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}$$