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Step by step

1. \int_{- \pi}^{\pi} \sin^{2}{\left(x \right)}\, dx

   First find an antiderivative F, then evaluate F(b) − F(a).

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

   Rewrite the integrand into a friendlier form.

3. \int \frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}\, dx = \int \frac{1}{2}\, dx + \int - \frac{\cos{\left(2 x \right)}}{2}\, dx

   The integral of a sum is the sum of the integrals.

4. \int \frac{1}{2}\, dx = \frac{x}{2}

   The integral of a constant c is c·x.

5. \int - \frac{\cos{\left(2 x \right)}}{2}\, dx = - \frac{1}{2} \int \cos{\left(2 x \right)}\, dx

   Pull the constant - \frac{1}{2} out of the integral.

6. u = 2 x,\quad du = 2\, dx

   Substitute u = 2 x.

7. \int \cos{\left(2 x \right)}\, dx = \int \frac{\cos{\left(u \right)}}{2}\, d_u

   Rewrite the integral in terms of u.

8. \int \frac{\cos{\left(u \right)}}{2}\, d_u = \frac{1}{2} \int \cos{\left(u \right)}\, d_u

   Pull the constant \frac{1}{2} out of the integral.

9. \int \cos{\left(u \right)}\, d_u = \sin{\left(u \right)}

   Standard trigonometric antiderivative.

10. = \frac{\sin{\left(2 x \right)}}{2}

    Substitute back u = 2 x.

11. F(\pi) - F(- \pi) = \left(\frac{\pi}{2}\right) - \left(- \frac{\pi}{2}\right)

    Fundamental theorem of calculus: plug in the limits.

12. = \pi \approx 3.1416

    Simplify.