1) int dx/(x^(5/6) - x^(1/2)), 2) int (x^2 - a^2)^(1/2)/x dx, 3) int (tg^2 x + tg^4 x) dx Опции

[Игорь]

Помогите, пожалуйста, найти интеграл
int dx/(x^(5/6) - x^(1/2))
Заранее спасибо!

[tig81]

int dx/(x^(5/6) - x^(1/2)) = | x^(1/6) = t, x = t^6, dx = 6 * t^5 dt | =
= int 6 * t^5/(t^5 - t^3) dt = 6 * int t^5/(t^3 * (t^2 - 1)) dt =
= 6 * int t^2/(t^2 - 1) dt = 6 * int (t^2 - 1 + 1)/(t^2 - 1) dt =
= 6 * int dt + 6 * int dt/(t^2 - 1) = 6 * t + 3 * int 2/((t - 1) * (t + 1)) dt =
= 6 * t + 3 * int ((t + 1) - (t - 1))/((t - 1) * (t + 1)) dt =
= 6 * t + 3 * int dt/(t - 1) - 3 * int dt/(t + 1) =
= 6 * t + 3 * ln |t - 1| - 3 * ln |t + 1| + C = 6 * t + 3 * ln |(t - 1)/(t + 1)| + C =
= | t = x^(1/6) | = 6 * x^(1/6) + 3 * ln |(x^(1/6) - 1)/(x^(1/6) + 1)| + C

[Игорь]

Спасибо большое!!
Подскажите, пожалуста, с чего здесь начать:
int (x^2 - a^2)^(1/2)/x dx

[tig81]

int (x^2 - a^2)^(1/2)/x dx =
= | t = (x^2 - a^2)^(1/2), dt = 1/2 * (x^2 - a^2)^(-1/2) * 2 * x dx =>
=> dx = (x^2 - a^2)^(1/2)/x dt = t/x dt, t^2 = x^2 - a^2 | =
= int t/x * t/x dt = int t^2/x^2 dt = int t^2/(t^2 + a^2) dt =
= int (t^2 + a^2 - a^2)/(t^2 + a^2) dt =
= int dt - a^2 * int dt/(t^2 + a^2) = t - a^2 * 1/a * arctg (t/a) + C =
= | t = (x^2 - a^2)^(1/2) | =
= (x^2 - a^2)^(1/2) - a * arctg ((x^2 - a^2)^(1/2)/a) + C

[Игорь]

Спасибо большое (IMG:style_emoticons/default/bigwink.gif)

[Игорь]

Помогите, пожалуйста, найти интеграл
int (tg^2 x + tg^4 x) dx

[tig81]

int (tg^2 x + tg^4 x) dx = int (tg^2 x * (1 + tg^2 x)) dx =
= int (tg^2 x * (1 + sin^2 x/cos^2 x)) dx =
= int (tg^2 x * 1/cos^2 x) dx = int tg^2 x d(tg x) = | t = tg x | =
= int t^2 dt = 1/3 * t^3 + C = | t = tg x | = 1/3 * tg^3 x + C

[Игорь]

Спасибо большое!! (IMG:style_emoticons/default/smile.gif)