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Консультация № 161271
Обсуждение
Неизвестный
25.02.2009, 11:00
общий
это ответ
Здравствуйте, Vanekkk!
2) Int[(10x-14)dx/(x3-6x2-7x)]=(*)
(10x-14)/(x3-6x2-7x)=(10x-14)/(x(x-7)(x+1))=A/x + B/(x-7) + C/(x+1) = (A(x-7)(x+1)+Bx(x+1)+Cx(x-7))/(x(x-7)(x+1))
10x-14=A(x-7)(x+1)+Bx(x+1)+Cx(x-7)
10x-14=Ax2-6Ax-7A+Bx2+Bx+Cx2-7Cx
0=A+B
10=-6A+B-7C
-14=-7A
B=-2
C=-24/7
A=2
(10x-14)/(x3-6x2-7x)=A/x + B/(x-7) + C/(x+1) =2/x - 2/(x-7) - 24/(7*(x+1))
(*)=Int[(2/x - 2/(x-7) - 24/(7*(x+1)))dx]=2*Int[dx/x]-2Int[dx/(x-7)]-24/7*Int[dx/(x+1)]=2*lnx-2*ln(x-7)-24/7*ln(x+1)+C
3) Int[dx/(x(x-1)3)]=(**)
1/(x(x-1)3)=A/x + B/(x-1) + C/(x-1)2 + D/(x-1)3=(A(x-1)3+Bx(x-1)2+Cx(x-1)+Dx)/(x(x-1)3)
1=Ax3-3Ax2+3Ax-A+Bx3-2Bx2+Bx+Cx2-Cx+Dx
1=x3(A+B)+x2(-3A-2B+C)+x(3A+B-C+D)+(-A)
0=A+B
0=-3A-2B+C
0=3A+B-C+D
1=-A
B=1
C=-1
D=1
A=-1
1/(x(x-1)3)=A/x + B/(x-1) + C/(x-1)2 + D/(x-1)3=-1/x + 1/(x-1) - 1/(x-1)2 + 1/(x-1)3
(**)=Int[(-1/x + 1/(x-1) - 1/(x-1)2 + 1/(x-1)3)dx]=-Int[dx/x]+Int[dx/(x-1)]-Int[dx/(x-1)2]+Int[dx/(x-1)3]=
=-lnx+ln(x-1)+1/(x-1)-1/2*1/(x-1)2+C
2) Int[(10x-14)dx/(x3-6x2-7x)]=(*)
(10x-14)/(x3-6x2-7x)=(10x-14)/(x(x-7)(x+1))=A/x + B/(x-7) + C/(x+1) = (A(x-7)(x+1)+Bx(x+1)+Cx(x-7))/(x(x-7)(x+1))
10x-14=A(x-7)(x+1)+Bx(x+1)+Cx(x-7)
10x-14=Ax2-6Ax-7A+Bx2+Bx+Cx2-7Cx
0=A+B
10=-6A+B-7C
-14=-7A
B=-2
C=-24/7
A=2
(10x-14)/(x3-6x2-7x)=A/x + B/(x-7) + C/(x+1) =2/x - 2/(x-7) - 24/(7*(x+1))
(*)=Int[(2/x - 2/(x-7) - 24/(7*(x+1)))dx]=2*Int[dx/x]-2Int[dx/(x-7)]-24/7*Int[dx/(x+1)]=2*lnx-2*ln(x-7)-24/7*ln(x+1)+C
3) Int[dx/(x(x-1)3)]=(**)
1/(x(x-1)3)=A/x + B/(x-1) + C/(x-1)2 + D/(x-1)3=(A(x-1)3+Bx(x-1)2+Cx(x-1)+Dx)/(x(x-1)3)
1=Ax3-3Ax2+3Ax-A+Bx3-2Bx2+Bx+Cx2-Cx+Dx
1=x3(A+B)+x2(-3A-2B+C)+x(3A+B-C+D)+(-A)
0=A+B
0=-3A-2B+C
0=3A+B-C+D
1=-A
B=1
C=-1
D=1
A=-1
1/(x(x-1)3)=A/x + B/(x-1) + C/(x-1)2 + D/(x-1)3=-1/x + 1/(x-1) - 1/(x-1)2 + 1/(x-1)3
(**)=Int[(-1/x + 1/(x-1) - 1/(x-1)2 + 1/(x-1)3)dx]=-Int[dx/x]+Int[dx/(x-1)]-Int[dx/(x-1)2]+Int[dx/(x-1)3]=
=-lnx+ln(x-1)+1/(x-1)-1/2*1/(x-1)2+C
Форма ответа
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