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1. \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx

   Find an antiderivative.

2. u = \cos{\left(x \right)},\quad du = - \sin{\left(x \right)}\, dx

   Substitute u = \cos{\left(x \right)}.

3. \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx = \int - u\, d_u

   Rewrite the integral in terms of u.

4. \int - u\, d_u = -1 \int u\, d_u

   Pull the constant -1 out of the integral.

5. \int u\, d_u = \frac{u^{2}}{2}

   Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).

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

   Substitute back u = \cos{\left(x \right)}.

7. F(x) = - \frac{\cos^{2}{\left(x \right)}}{2} + C

   Add the constant of integration.