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Консультация № 158560
23.01.2009, 19:38
0.00 руб.
0
3
1
Здравствуйте, уважаемые эксперты, очень нужно решить интеграл, ничего не получается... помогите
[$8747$][от 1 до 2] ( (1/x^2)*sqrt(1+4x^2) )dx
Заранее спасибо.
[$8747$][от 1 до 2] ( (1/x^2)*sqrt(1+4x^2) )dx
Заранее спасибо.
Обсуждение
Неизвестный
23.01.2009, 20:16
общий
это ответ
Здравствуйте, Ingenio!
Здесь удобно сделать замену : 2*х=tgt , тогда dx=dt/(2*(cost)^2) и 1+4*x^2=1/((cost)^2) .
∫[от 1 до 2] ( (1/x^2)*sqrt(1+4x^2) )dx = INT[(4*cost*dt)/(2*((cost)^2)*((tgt)^2))] = 2*INT[cost*dt/((sint)^2)]=-2/sint
Переменная t изменяется от arctg2 до arctg4 , можно подсчитать на калькуляторе , а можно вспомнить что sint=tgt/sqrt(1+((tgt)^2)) и тогда получится без калькулятора , ведь tg(arctgx)=x ...
∫[от 1 до 2] ( (1/x^2)*sqrt(1+4x^2) )dx = -[2*sqrt(1+16)/4]+[2*sqrt(1+4)/2] = sqrt5-((sqrt17)/2) = 0,174515164 .
Здесь удобно сделать замену : 2*х=tgt , тогда dx=dt/(2*(cost)^2) и 1+4*x^2=1/((cost)^2) .
∫[от 1 до 2] ( (1/x^2)*sqrt(1+4x^2) )dx = INT[(4*cost*dt)/(2*((cost)^2)*((tgt)^2))] = 2*INT[cost*dt/((sint)^2)]=-2/sint
Переменная t изменяется от arctg2 до arctg4 , можно подсчитать на калькуляторе , а можно вспомнить что sint=tgt/sqrt(1+((tgt)^2)) и тогда получится без калькулятора , ведь tg(arctgx)=x ...
∫[от 1 до 2] ( (1/x^2)*sqrt(1+4x^2) )dx = -[2*sqrt(1+16)/4]+[2*sqrt(1+4)/2] = sqrt5-((sqrt17)/2) = 0,174515164 .
Неизвестный
24.01.2009, 12:10
общий
Спасибо большое, Айболит. Я правда не совсем понял, как путём подстановки в исходный интеграл замены, мы получили
INT[(4*cost*dt)/(2*((cost)^2)*((tgt)^2))] ? У меня получается, если подставить следующее:
INT[(2dt/((sint^2)*cost))]... и чему это дальше ровняется я не понимаю...
INT[(4*cost*dt)/(2*((cost)^2)*((tgt)^2))] ? У меня получается, если подставить следующее:
INT[(2dt/((sint^2)*cost))]... и чему это дальше ровняется я не понимаю...
Неизвестный
24.01.2009, 12:58
общий
Простите , неправильно понял условие .
INT[(2dt/((sint^2)*cost))]=2*INT[(cost*dt)/((cost*sint)^2)]=2*INT[d(sint)/((sint)^2)]+2*INT[d(sint)/(1-((sint)^2))]=
=-[2/sint]+Ln[(1+sint)/(1-sint)]=...
Ну , новые пределы интегрирования , то есть синус t изменяются от 4/sqrt17 до 2/sqrt5
INT[(2dt/((sint^2)*cost))]=2*INT[(cost*dt)/((cost*sint)^2)]=2*INT[d(sint)/((sint)^2)]+2*INT[d(sint)/(1-((sint)^2))]=
=-[2/sint]+Ln[(1+sint)/(1-sint)]=...
Ну , новые пределы интегрирования , то есть синус t изменяются от 4/sqrt17 до 2/sqrt5
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