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Консультация № 160922
19.02.2009, 19:50
0.00 руб.
0
1
1
Здравствуйте эксперты!
Помогите взять интеграл, неделю ломаю голову...
dx
---------------------------
(2 - 6x - x^2) ^ (5/2)
Заранее большое спасибо!
Помогите взять интеграл, неделю ломаю голову...
dx
---------------------------
(2 - 6x - x^2) ^ (5/2)
Заранее большое спасибо!
Обсуждение
Неизвестный
19.02.2009, 20:23
общий
это ответ
Здравствуйте, Tossha!
Выражение в знаменателе немного преобразуем .
(2 - 6x - x^2) ^ (5/2)=(2+9-((x^2)+6x+9))^(5/2)=(11-((x+3)^2))^(5/2)
Делаем замену : x+3=(sqrt(11))*sint , dx=(sqrt(11))*cost*dt .
В знаменателе выносим 11 за корень . Как известно 1-((sint)^2)=(cost)^2 .
INT[dx/((2 - 6x - x^2) ^ (5/2))]=INT[(sqrt11)*cost*dt/(121*(sqrt11)*((cost)^5))]=(1/121)*INT[dt/((cost)^4)]=
=(1/121)*INT[(((tgt)^2)+1)*(dt/((cost)^2))]=(1/121)*INT[(((tgt)^2)+1)*d(tgt)]=(1/363)*((tgt)^3)+(1/121)*tgt+C .
Вспомним что tgt=sint/sqrt(1-((sint)^2)) .
sint=(x+3)/sqrt11 => tgt=((sqrt11)/11)*((x+3)^2)/sqrt(11-((x+3)^2))=(1/sqrt11)*((x+3)^2)/sqrt(2-6x-(x^2))=tgt .
Ответ в тангенсах надо обязательно выразить через х .
Выражение в знаменателе немного преобразуем .
(2 - 6x - x^2) ^ (5/2)=(2+9-((x^2)+6x+9))^(5/2)=(11-((x+3)^2))^(5/2)
Делаем замену : x+3=(sqrt(11))*sint , dx=(sqrt(11))*cost*dt .
В знаменателе выносим 11 за корень . Как известно 1-((sint)^2)=(cost)^2 .
INT[dx/((2 - 6x - x^2) ^ (5/2))]=INT[(sqrt11)*cost*dt/(121*(sqrt11)*((cost)^5))]=(1/121)*INT[dt/((cost)^4)]=
=(1/121)*INT[(((tgt)^2)+1)*(dt/((cost)^2))]=(1/121)*INT[(((tgt)^2)+1)*d(tgt)]=(1/363)*((tgt)^3)+(1/121)*tgt+C .
Вспомним что tgt=sint/sqrt(1-((sint)^2)) .
sint=(x+3)/sqrt11 => tgt=((sqrt11)/11)*((x+3)^2)/sqrt(11-((x+3)^2))=(1/sqrt11)*((x+3)^2)/sqrt(2-6x-(x^2))=tgt .
Ответ в тангенсах надо обязательно выразить через х .
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