سیگما

`S=int (x^4-2x^2+2)/((x^2-2x+2)^2)dx`
`S=int1+(2(x-1)^2(2x-1))/((x^2-2x+2)^2)dx`
`S=x+T`
`T=int (2(x-1)^2(2x-1))/((x^2-2x+2)^2)dx`
`T=((P_1)(x))/((Q_1)(x))+int ((P_2)(x))/((Q_2)(x))dx`
`(Q_1)(x)=x^2-2x+2`
`(Q_2)(x)=x^2-2x+2`
`T=(Ax+B)/(x^2-2x+2)+int(Cx+D)/(x^2-2x+2)dx`
`A=-1 , B=3 , C=4 , D=-3`
`T=(-x+3)/(x^2-2x+2)+int(4x-3)/(x^2-2x+2)dx`
`T=(-x+3)/(x^2-2x+2)+2int(2x-2+(2-3/2))/(x^2-2x+2)dx`
`T=(-x+3)/(x^2-2x+2)+2(ln(x^2-2x+2)+1/2int(dx)/(x^2-2x+2))`
`T=(-x+3)/(x^2-2x+2)+2(ln(x^2-2x+2)+1/2int(dx)/((x-1)^2+1))`
`T=(-x+3)/(x^2-2x+2)+2ln|x^2-2x+2|+arctan(x+1)`
`S=x+(-x+3)/(x^2-2x+2)+2ln(x^2-2x+2)+arctan(x-1)`

`S=int (dx)/((x+1)^2(x^2+1)^2)`
`S=((P_1)(x))/((Q_1)(x))+int ((P_2)(x))/((Q_2)(x))dx`
`Q(x)=(x+1)^2(x^2+1)^2`
`(Q_1)(x)=(x+1)(x^2+1)`
`(Q_2)(x)=(x+1)(x^2+1)`
`S=(Ax^2+Bx+C)/((x+1)(x^2+1))+int(Dx^2+Ex+F)/((x+1)(x^2+1))dx`
`A=-1/4 , B=1/4 , C=0 , D=0 , E=-1/4 , F=3/4`
`S=(-1/4x^2+1/4x)/((x+1)(x^2+1))+int(-1/4x+3/4)/((x+1)(x^2+1))dx`
`S=(-1/4x^2+1/4x)/(x+1)(x^2+1))+T`
`T=int(-1/4x+3/4)/(x+1)(x^2+1))dx`
`T=1/4int(3-x)/((x+1)(x^2+1))dx=1/4intA/(x+1)+(Bx+C)/(x^2+1)dx`
`A=2 , B=-2 , C=1`
`T=1/2ln|x+1|+1/4int(-2x+1)/(x^2+1)dx`
`T=1/2ln|x+1|-1/4int(2x-1)/(x^2+1)dx`
`T=(ln|x+1|)/2-1/4(ln|x^2+1|-int(dx)/(x^2+1))`
`T=(ln|(x+1)^2/(x^2+1)|+arctanx)/4`
`S=((x-x^2)/((x+1)(x^2+1))+ln|(x+1)^2/(x^2+1)|+arctanx)/4`

`Q=int 1/(x^4+1)dx`
`Q=int 1/((x^2-sqrt(2)x+1)(x^2+sqrt(2)x+1))dx`
`Q=int (Ax+B)/(x^2-sqrt(2)x+1)+(Cx+D)/(x^2+sqrt(2)x+1)dx`
`Q=sqrt(2)/8(int (2x+2sqrt(2))/(x^2+sqrt(2)x+1)dx-int (2x-2sqrt(2))/(x^2-sqrt(2)x+1)dx)`
`Q=sqrt(2)/8(ln|(x^2+sqrt(2)x+1)/(x^2-sqrt(2)x+1)|+sqrt(2)int (dx)/(x^+sqrt(2)x+1)-sqrt(2)int (dx)/(x^2-sqrt(2)x+1))`
`Q=sqrt(2)/8(ln|(x^2+sqrt(2)x+1)/(x^2-sqrt(2)x+1)|+sqrt(2)int (dx)/((x+sqrt(2)/2)^2+1/2)-sqrt(2)int (dx)/((x-sqrt(2)/2)^2+1/2))`
`Q=sqrt(2)/8(ln|(x^2+sqrt(2)x+1)/(x^2-sqrt(2)x+1)|+2arctan(1+sqrt(2)x)-2arctan(1-sqrt(2)x))`

`Q=int (dx)/(x^3+1)`
`Q=int (dx)/((x+1)(x^2-x+1))`
`Q=int A/(x+1)+(Bx+C)/(x^2-x+1)dx`
`A=1/3 , B=-1/3 , C=2/3`
`Q=1/3 int 1/(x+1)dx+int (-1/3x+2/3)/(x^2-x+1)dx`
`Q=1/3ln(x+1)-1/6int(2x-1-3)/(x^2-x+1)dx`
`Q=2/6ln(x+1)-1/6ln(x^2-x+1)+1/2int(dx)/(x^2-x+1)dx`
`Q=1/6ln|(x+1)^2|-1/6ln(x^2-x+1)+1/2int(dx)/((x-1/2)^2+3/4)dx`
`Q=(ln|(x+1)^2/(x^2-x+1)|)/6+1/sqrt(3)arctan((2x-1)/(sqrt(3)))`

`Q=int (x^4)/(x^4-1)dx`
`Q=int 1+1/(x^4-1)dx`
`Q=x+P`
`P=int 1/(x^4-1)dx=int1/((x-1)(x+1)(x^2+1))dx=int A/(x-1)+B/(x+1)+(Cx+D)/(x^2+1)dx`
`A=1/4 , B=-1/4 , C=0 , D=-1/2`
`P=int (1/4)/(x-1)-(1/4)/(x+1)-(1/2)/(x^2+1)dx`
`Q=x+1/4ln|(x-1)/(x+1)|-1/2arctanx`