سیگما

`Q=int (dx)/(sqrt(x+1)+sqrt((x+1)^3))`
`t^2=x+1 , dx= 2tdt`
`Q=int (2t)/(t+t^3)dt`
`Q=2 int (dt)/(t^2+1)`
`Q=2arctan(t)`
`Q=2arctan(sqrt(x+1))`

`Q=int(dx)/((x+1)^5sqrt(x^2+2x))`
`1/t=x+1`
`x=1/t-1`
`(dx)/(dt)=-1/(t^2)`
`Q=int (-1/(t^2))/((1/t)^5 sqrt((1/t-1)^2+2(1/t-1))) dt`
`Q=-int (t^3)/(sqrt((1/t)^2-1)) dt`
`Q=int (-t^4)/sqrt(1-t^2) dt`
`Q=(At^3+Bt^2+Ct+D)sqrt(1-t^2)+ K int 1/(sqrt(1-t^2)) dt`
`A=1/4,B=0,C=3/8,D=0,K=-3/8`
`Q=(1/4t^3+3/8t)sqrt(1-t^2)-3/8arcsin(t)`
`Q=1/8(t(2t^2+3)sqrt(1-t^2)-3arcsint)`
`|t=(1/(x+1))`

`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|`