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Консультация № 109229
13.11.2007, 15:54
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! 4х arc cos x dx
! - это знак интеграла.
Помогите пожалуйста кто нибудь! Буду презнателен!
! - это знак интеграла.
Помогите пожалуйста кто нибудь! Буду презнателен!
Обсуждение
Неизвестный
13.11.2007, 17:25
общий
это ответ
Здравствуйте, Caspersurgut!
берем по частям
u=arccosx dv=xdx du=-dx/√(1-x^2) v=x^2/2
тогда,
∫4х arc cos x dx= 4(arccosx*x^2/2-∫-(x^2)dx/2√(1-x^2) )=
чтобы посчитать этот интеграл заменим x=cosy. dx=-sinydy
∫-(x^2)dx/2√(1-x^2) )=∫(cos^2y siny dy)/2siny=1/4*∫(1+cos2y)dy=(1/4)*y+(1/8)sin2y=
=1/4 arccosx + 1/8 sin(2arccosx)
Вернемся к интегралу
=2(arccosx)*(x^2) - 4*(1/4 arccosx + 1/8 sin(2arccosx))+c =
{sin(2arccosx)=2√(1-cos^2 (arccosx))*cos(arccosx)= 2x√(1-x^2)}
= 2(arccosx)*(x^2) - arccosx - 1/2 * 2x√(1-x^2) + c =
= 2(arccosx)*(x^2) - arccosx - x*√(1-x^2) + c
<p><fieldset style=‘background-color:#EFEFEF; width:80%; border:#777777 1px solid; padding:10px;‘ class=fieldset><font color=#777777><i>Исправлено и дополнено по просьбе эксперта</i>
-----
</font><font color=#777777 size=1><b>• Отредактировал: <a href=/info/user/118729 target=_blank>Агапов Марсель</a></b> (Специалист)
<b>• Дата редактирования:</b> 13.11.2007, 19:36</font></fieldset>
берем по частям
u=arccosx dv=xdx du=-dx/√(1-x^2) v=x^2/2
тогда,
∫4х arc cos x dx= 4(arccosx*x^2/2-∫-(x^2)dx/2√(1-x^2) )=
чтобы посчитать этот интеграл заменим x=cosy. dx=-sinydy
∫-(x^2)dx/2√(1-x^2) )=∫(cos^2y siny dy)/2siny=1/4*∫(1+cos2y)dy=(1/4)*y+(1/8)sin2y=
=1/4 arccosx + 1/8 sin(2arccosx)
Вернемся к интегралу
=2(arccosx)*(x^2) - 4*(1/4 arccosx + 1/8 sin(2arccosx))+c =
{sin(2arccosx)=2√(1-cos^2 (arccosx))*cos(arccosx)= 2x√(1-x^2)}
= 2(arccosx)*(x^2) - arccosx - 1/2 * 2x√(1-x^2) + c =
= 2(arccosx)*(x^2) - arccosx - x*√(1-x^2) + c
<p><fieldset style=‘background-color:#EFEFEF; width:80%; border:#777777 1px solid; padding:10px;‘ class=fieldset><font color=#777777><i>Исправлено и дополнено по просьбе эксперта</i>
-----
</font><font color=#777777 size=1><b>• Отредактировал: <a href=/info/user/118729 target=_blank>Агапов Марсель</a></b> (Специалист)
<b>• Дата редактирования:</b> 13.11.2007, 19:36</font></fieldset>
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