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Консультация № 109252
13.11.2007, 17:40
0.00 руб.
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Интеграл:
7: S (7x-15/x^3-2x^2+5x)*dx
Приложение:
S - условное обозначение интеграла.
7: S (7x-15/x^3-2x^2+5x)*dx
Приложение:
S - условное обозначение интеграла.
Обсуждение
Неизвестный
13.11.2007, 19:00
общий
это ответ
Здравствуйте, Packbar!
∫(7x-15/x^3-2x^2+5x)*dx
разобьем дробь в сумму дробей
a/x + (bx+c)/(x^2-2x+5) найдем коэффициенты a, в с
ax^2-2ax+5a+bx^2+cx=7x-15
a+b=0, -2a+c=7, 5a=-15
a=-3, b=3, c=1
итак, ∫(7x-15/x^3-2x^2+5x)*dx =∫-3dx/x +3∫xdx/(x^2-2x+5)+
+∫dx/(x^2-2x+5)
∫xdx/(x^2-2x+5)=∫(x-1+1)dx/(x^2-2x+5)=
=1/2∫(2x-2)dx/(x^2-2x+5)+∫dx/(x^2-2x+5)=1/2ln(x^2-2x+5)+∫dx/(x-1)^2+4)=
=(1/2)ln(x^2-2x+5)+(1/2)arctg(x/2)+с
вернемся к исходному интегралу:
∫(7x-15/x^3-2x^2+5x)*dx =∫-3dx/x +3∫xdx/(x^2-2x+5)+∫dx/(x^2-2x+5)=
=-3lnx+(3/2)ln(x^2-2x+5)+(3/2)arctg(x/2)+(1/2)arctg(x/2)+с=
=-3lnx+(3/2)ln(x^2-2x+5)+2arctg(x/2)+с
∫(7x-15/x^3-2x^2+5x)*dx
разобьем дробь в сумму дробей
a/x + (bx+c)/(x^2-2x+5) найдем коэффициенты a, в с
ax^2-2ax+5a+bx^2+cx=7x-15
a+b=0, -2a+c=7, 5a=-15
a=-3, b=3, c=1
итак, ∫(7x-15/x^3-2x^2+5x)*dx =∫-3dx/x +3∫xdx/(x^2-2x+5)+
+∫dx/(x^2-2x+5)
∫xdx/(x^2-2x+5)=∫(x-1+1)dx/(x^2-2x+5)=
=1/2∫(2x-2)dx/(x^2-2x+5)+∫dx/(x^2-2x+5)=1/2ln(x^2-2x+5)+∫dx/(x-1)^2+4)=
=(1/2)ln(x^2-2x+5)+(1/2)arctg(x/2)+с
вернемся к исходному интегралу:
∫(7x-15/x^3-2x^2+5x)*dx =∫-3dx/x +3∫xdx/(x^2-2x+5)+∫dx/(x^2-2x+5)=
=-3lnx+(3/2)ln(x^2-2x+5)+(3/2)arctg(x/2)+(1/2)arctg(x/2)+с=
=-3lnx+(3/2)ln(x^2-2x+5)+2arctg(x/2)+с
Форма ответа
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