I = int cos 2x/e^x dx = int cos 2x * e^(-x) dx = -int cos 2x d(e^(-x)) =
= -cos 2x * e^(-x) + int e^(-x) d(cos 2x) =
= -cos 2x * e^(-x) - 2 * int e^(-x) * sin 2x dx =
= -cos 2x * e^(-x) + 2 * int sin 2x d(e^(-x)) =
= -cos 2x * e^(-x) + 2 * sin 2x * e^(-x) - 2 * int e^(-x) d(sin 2x) =
= -cos 2x * e^(-x) + 2 * sin 2x * e^(-x) - 4 * int cos 2x * e^(-x) dx + C
Получили, что
I = -cos 2x * e^(-x) + 2 * sin 2x * e^(-x) + C - 4 * I
5 * I = -cos 2x * e^(-x) + 2 * sin 2x * e^(-x) + C
Значит
int cos 2x/e^x dx = -1/5 * cos 2x * e^(-x) + 2/5 * sin 2x * e^(-x) + C