Example 1 (kp. $1^{\circ}$ ). Find $\int \sin 5 x \cos 7 x d x$.
Solution. We have

$\int \sin 5 x \cos 7 x d x=\frac{1}{2} \int(-\sin 2 x+\sin 12 x) d x=\frac{1}{4} \cos 2 x-\frac{1}{24} \cos 12 x+C$.