سیگما

`Q=int (dx)/(x-sqrt(x^2+2x+4))`
`sqrt(x^2+2x+4)=t+x`
`x=(t^2-4)/(2-2t)`
`(dx)/(dt)=(-t^2+2t-4)/(2(1-t)^2)`
`Q=int ((-t^2+2t-4)/(2(1-t)^2))/(-t) dt`
`Q=int (t^2-2t+4)/(2t(1-t)^2) dt`
`Q=2/t-(3/2)/(t-1)+(3/2)/(t-1)^2`
`Q=2lnt-3/2ln(t-1)-3/(2(t-1))`
`|t=(sqrt(x^2+2x+4)-x)`

`Q=int root3((x+1)/(x-1)) dx`
`t^3=(x+1)/(x-1)`
`x=(t^3+1)/(t^3-1)`
`(dx)/(dt)=(-6t^2)/(t^3-1)^2`
`Q=-int (6t^3)/(t^3-1)^2 dt`
`(6t^3)/(t^3-1)^2=A/(t-1)+B/(t-1)^2+(Cx+D)/(t^2+t+1)+(Ex+F)/(t^2+t+1)^2`
`A=2/3,B=2/3,C=-2/3,D=-2,E=2,F=2`
`Q=-int (2t+2)/(t^2+t+1)^2-(2t+6)/(3(t^2+t+1))+2/(3(t-1))+2/(3(t-1)^2) dt`
`Q=1/3((6t)/(t^3-1)+ln((t^2+t+1)/(1-t)^2)+2sqrt(3)arctan((2t+1)/sqrt(3)))`
`t=((x+1)/(x-1))^1/3`

`int (x^4+x^2+1)/(x-1) dx`
`int x^3+x^2+2x+2+3/(x-1) dx`
`x^4/4+x^3/3+x^2+2x+3ln|x-1|`

`int (dx)/(sqrt(4+x^2))`
`int (dx)/(sqrt(2^2+x^2))`
`arcsinh(x/a)`
`ln|x+sqrt(4+x^2)|`

`int (aqrt(a)-sqrt(x))^4/(sqrt(ax)) dx`
`int (a-2sqrt(ax)+x)^2/(sqrt(ax)) dx`
`int (a^2+6ax-4asqrt(ax)-4xsqrt(ax)+x^2)/(sqrt(ax)) dx`
`int a sqrt(a)/sqrt(x)+6sqrt(ax)-4a-4x+sqrt(x^3)/sqrt(a) dx`
`2a sqrt(ax)-4ax+4x sqrt(ax)-2x^2+(2x^3)/(5sqrt(ax))`

`int (a+bx^3)^2 dx`
`int a^2+2abx^3+b^2x^6 dx`
`a^2x+(2abx^4)/4+(b^2x^7)/7`

`int x(x+a)(x+b) dx`
`int x(x^2+(a+b)x+ab) dx`
`int x^3+(a+b)x^2+abx dx`
`(x^4)/4+((a+b)x^3)/3+(abx^2)/2`

`Q=int (x^3)/sqrt(x-1)dx`
`x-1=t^2`
`dx=2tdt`
`x=t^2+1`
`Q=int((t^2+1)^3/t)2t dt`
`Q=2int(t^2+1)^3dt`
`Q=2int t^6+3t^4+3t^2+1 dt`
`Q=2sqrt(x-1)(1/7(x-1)^3+3/5(x-1)^2+x)`

`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`