2/((x^4-1)x)=2/((x^2-1)(x^2+1)x)=2/((x-1)(x+1)(x^2+1)x)2(x4−1)x=2(x2−1)(x2+1)x=2(x−1)(x+1)(x2+1)x
2/((x-1)(x+1)(x^2+1)x)=A/x+B/(x-1)+C/(x+1)+(Dx+E)/(x^2+1)=2(x−1)(x+1)(x2+1)x=Ax+Bx−1+Cx+1+Dx+Ex2+1=
=(A(x-1)(x+1)(x^2+1))/(x(x-1)(x+1)(x^2+1))+(Bx(x+1)(x^2+1))/(x(x-1)(x+1)(x^2+1))+=A(x−1)(x+1)(x2+1)x(x−1)(x+1)(x2+1)+Bx(x+1)(x2+1)x(x−1)(x+1)(x2+1)+
+(Cx(x-1)(x^2+1))/(x(x-1)(x+1)(x^2+1))+((Dx+E)x(x-1)(x+1))/(x(x-1)(x+1)(x^2+1))=+Cx(x−1)(x2+1)x(x−1)(x+1)(x2+1)+(Dx+E)x(x−1)(x+1)x(x−1)(x+1)(x2+1)=
=(A(x^4-1))/(x(x-1)(x+1)(x^2+1))+(Bx(x^3+x^2+x+1))/(x(x-1)(x+1)(x^2+1))+=A(x4−1)x(x−1)(x+1)(x2+1)+Bx(x3+x2+x+1)x(x−1)(x+1)(x2+1)+
+(Cx(x^3-x^2+x-1))/(x(x-1)(x+1)(x^2+1))+((Dx+E)(x^3-x))/(x(x-1)(x+1)(x^2+1))=+Cx(x3−x2+x−1)x(x−1)(x+1)(x2+1)+(Dx+E)(x3−x)x(x−1)(x+1)(x2+1)=
=(Ax^4-A)/(x(x-1)(x+1)(x^2+1))+(Bx^4+Bx^3+Bx^2+Bx)/(x(x-1)(x+1)(x^2+1))+=Ax4−Ax(x−1)(x+1)(x2+1)+Bx4+Bx3+Bx2+Bxx(x−1)(x+1)(x2+1)+
+(Cx^4-Cx^3+Cx^2-Cx)/(x(x-1)(x+1)(x^2+1))+(Dx^4+Ex^3-Dx^2-Ex)/(x(x-1)(x+1)(x^2+1))=+Cx4−Cx3+Cx2−Cxx(x−1)(x+1)(x2+1)+Dx4+Ex3−Dx2−Exx(x−1)(x+1)(x2+1)=
=(x^4(A+B+C+D)+x^3(B-C+E))/(x(x-1)(x+1)(x^2+1))+=x4(A+B+C+D)+x3(B−C+E)x(x−1)(x+1)(x2+1)+
+(x^2(B+C-D)+x(B-C-E)+(-A))/(x(x-1)(x+1)(x^2+1))+x2(B+C−D)+x(B−C−E)+(−A)x(x−1)(x+1)(x2+1)
A+B+C+D=0A+B+C+D=0
B-C+E=0B−C+E=0
B+C-D=0B+C−D=0
B-C-E=0B−C−E=0
-A=2 => A=-2−A=2⇒A=−2
[II-IV] => 2E=0 => E=0[II−IV]⇒2E=0⇒E=0
B+C+D=2B+C+D=2
B-C=0 => B=CB−C=0⇒B=C
B+C-D=0B+C−D=0
2C+D=22C+D=2
2C-D=02C−D=0
4C=2 => C=1/2 => B=1/24C=2⇒C=12⇒B=12
D=2C => D=1D=2C⇒D=1
I=int 2/((x^4-1)x)dx= -2int dx/x +1/2int dx/(x-1) + 1/2int dx/(x+1) + int (xdx)/(x^2+1)I=∫2(x4−1)xdx=−2∫dxx+12∫dxx−1+12∫dxx+1+∫xdxx2+1
I=-2ln|x|+1/2ln|x-1|+1/2ln|x+1|+1/2ln|x^2+1|+CI=−2ln|x|+12ln|x−1|+12ln|x+1|+12ln∣∣x2+1∣∣+C
Note:
int (xdx)/(x^2+1) = int (1/2(2xdx))/(x^2+1) = 1/2int (d(x^2+1))/(x^2+1) = 1/2ln|x^2+1|∫xdxx2+1=∫12(2xdx)x2+1=12∫d(x2+1)x2+1=12ln∣∣x2+1∣∣
