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Консультация № 101201
07.09.2007, 15:38
0.00 руб.
0
2
2
Помогите решить интеграл sqrt(x*x-4)/x по dx или подскажите, в какую сторону копать. Пробовал замену x=2/sin(t), но потом получается интеграл ctg^2(t) по dt - не знаю что с этим делать.
Обсуждение
Неизвестный
07.09.2007, 16:06
общий
это ответ
Здравствуйте, Shb!
Попробуем замену t^2=x^2-4. Тогда x^2=t^2+4 и
2t*dt = 2x*dx = 2x^2*dx/x = 2(t^2+4)*dx/x,
т.е. dx/x = t*dt/(t^2+4).
Интеграл примет вид
integral(t*t*dt/(t^2+4)) = integral(t^2/(t^2+4)*dt) = integral(dt) - 4*integral(dt/(t^2+4)) = t - 4*1/2*arctg(t/2) + C = t - 2*arctg(t/2) + C.
Перейдём обратно к переменной x и получим
integral(sqrt(x^2-4)/x*dx) = sqrt(x^2-4) - 2*arctg(sqrt(x^2-4)/2).
Попробуем замену t^2=x^2-4. Тогда x^2=t^2+4 и
2t*dt = 2x*dx = 2x^2*dx/x = 2(t^2+4)*dx/x,
т.е. dx/x = t*dt/(t^2+4).
Интеграл примет вид
integral(t*t*dt/(t^2+4)) = integral(t^2/(t^2+4)*dt) = integral(dt) - 4*integral(dt/(t^2+4)) = t - 4*1/2*arctg(t/2) + C = t - 2*arctg(t/2) + C.
Перейдём обратно к переменной x и получим
integral(sqrt(x^2-4)/x*dx) = sqrt(x^2-4) - 2*arctg(sqrt(x^2-4)/2).
Форма ответа
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