Помогите, пожалуйста, найти эти два интеграла:
1) int dx/(x + 10)^3
2) int dx/(x^2 + 2)^2
Мне кажется, что здесь нужна какая-то замена.
Заранее спасибо.

1) int dx/(x + 10)^3 = int d(x + 10)/(x + 10)^3 = | t = x + 10 | =
= int dt/t^3 = int t^(-3) dt = 1/(-3 + 1) * t^(-3 + 1) + C = -1/2 * 1/t^2 + C =
= | t = x + 10 | = -1/2 * 1/(x + 10)^2 + C
2) int dx/(x^2 + 2)^2 = | x = 2^(1/2) * tg t; dx = 2^(1/2) * 1/cos^2 t dt | =
= int (2^(1/2) * 1/cos^2 t)/(2 * tg^2 t + 2)^2 dt =
= 2^(1/2) * int (1/cos^2 t)/(2^2 * (tg^2 t + 1)^2) dt =
= 2^(1/2)/4 * int (1/cos^2 t)/(sin^2 t/cos^2 t + 1)^2 dt =
= 2^(1/2)/4 * int (1/cos^2 t)/(1/cos^2 t)^2 dt =
= 2^(1/2)/4 * int (1/cos^2 t)/(1/cos^4 t) dt =
= 2^(1/2)/4 * int cos^4 t/cos^2 t dt = 2^(1/2)/4 * int cos^2 t dt =
= 2^(1/2)/4 * int (1 + cos (2t))/2 dt = 2^(1/2)/4 * int (1/2 + 1/2 * cos (2t)) dt =
= 2^(1/2)/4 * (1/2 * t + 1/2 * 1/2 * sin (2t)) + C =
= 2^(1/2)/4 * (1/2 * t + 1/4 * sin (2t)) + C =
= 2^(1/2)/8 * t + 2^(1/2)/16 * sin (2t) + C = | t = arctg (x/2^(1/2)) | =
= 2^(1/2)/8 * arctg (x/2^(1/2)) + 2^(1/2)/16 * sin (2 * arctg (x/2^(1/2))) + C
Вычислим sin (2 * arctg (x/2^(1/2))).
Пусть a = arctg x/2^(1/2) => tg a = x/2^(1/2)
tg^2 a + 1 = 1/cos^2 a => 1/cos^2 a = x^2/2 + 1 = (x^2 + 2)/2
cos^2 a = 2/(x^2 + 2)
Получаем, что
sin (2a) = 2 * sin a * cos a = 2 * sin a * cos^2 a/cos a =
= 2 * cos^2 a * sin a/cos a = 2 * cos^2 a * tg a =
= 2 * 2/(x^2 + 2) * x/2^(1/2) = 2 * 2^(1/2) * x/(x^2 + 2).
Тогда
int dx/(x^2 + 2)^2 =
= 2^(1/2)/8 * arctg (x/2^(1/2)) + 2^(1/2)/16 * 2 * 2^(1/2) * x/(x^2 + 2) + C =
= 2^(1/2)/8 * arctg (x/2^(1/2)) + 1/4 * x/(x^2 + 2) + C

Либо можно было проще
int dx/(x^2 + 2)^2 = 1/2 * int 2 dx/(x^2 + 2)^2 =
= 1/2 * int ((x^2 + 2) - x^2)/(x^2 + 2)^2 dx =
= 1/2 * int 1/(x^2 + 2) - 1/2 * int x^2/(x^2 + 2)^2 dx =
= 1/2 * 1/2^(1/2) * arctg (x/2^(1/2)) + 1/2 * int x/2 d(1/(2 + x^2)) =
= 1/2^(3/2) * arctg (x/2^(1/2)) + 1/4 * x/(x^2 + 2) - 1/2 * int 1/(x^2 + 2) d(x/2) =
= 1/2^(3/2) * arctg (x/2^(1/2)) + 1/4 * x/(x^2 + 2) - 1/4 * int 1/(x^2 + 2) dx =
= 1/2^(3/2) * arctg (x/2^(1/2)) + 1/4 * x/(x^2 + 2) -
- 1/4 * 1/2^(1/2) * arctg (x/2^(1/2)) + C =
= 1/2^(3/2) * arctg (x/2^(1/2)) + 1/4 * x/(x^2 + 2) -1/2^(5/2) * arctg (x/2^(1/2)) + C =
= 1/2^(5/2) * arctg (x/2^(1/2)) + 1/4 * x/(x^2 + 2) + C =
= 2^(1/2)/8 * arctg (x/2^(1/2)) + 1/4 * x/(x^2 + 2) + C

Спасибо за помощь!