سیگما

`Q=int root3(1+root4(x))/sqrt(x) dx`
`x=t^4`
`(dx)/(dt)=4t^3`
`Q=int root3(1+t)/t^2 4t^3 dt`
`t+1=u`
`du=dt`
`Q=4 int (u-1)root3(u) du`
`Q=4 int u^(4/3)-u^(1/3) du`
`Q=4(3/7u^(7/3)-3/4u^(4/3))`
`Q=12/7root3((1+root4(x))^7)-3root3((1+root4(x))^4)`

`Q=int (x^6)/sqrt(1+x^2) dx`
`Q=(Ax^5+Bx^4+Cx^3+Dx^2+Ex+F)sqrt(1+x^2)+K int (dx)/sqrt(1+x^2)`
`A=1/6,B=0,C=-5/24,D=0,E=5/16,F=0,K=-5/16`
`Q=(1/6x^5-5/24x^3+5/16x)sqrt(1+x^2)-5/16arcsinh(x)`

`Q=int (dx)/((x-1)sqrt(x^2-2))`
`t=1/(x-1)`
`x=1/t+1`
`(dx)/(dt)=-1/t^2`
`Q=int (-1/t)/sqrt(1/t^2+2/t+1-2) dt`
`Q=int (-1/t)/sqrt(1/t^2(1+2t-t^2)) dt`
`Q=-int (dt)/sqrt(2-(t-1)^2)`
`Q=-arcsin((t-1)/sqrt(2))`
`Q=arcsin((2-x)/(sqrt(2)(1-x)))`

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

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

`Q=int sin(x)sin(2x)sin(3x) dx`
`Q=1/2 int (2sin(2x)sin(x))sin(3x) dx`
`Q=1/2 int (cos(2x-x)-cos(2x+x))sin(3x) dx`
`Q=1/2 int (cos(x)-cos(3x))sin(3x) dx`
`Q=1/4 int 3sin(3x)cos(x)-2sin(3x)cos(3x) dx`
`Q=1/4 int (sin(3x+x)+sin(3x-x))-sin(6x) dx`
`Q=1/4 int sin(4x)+sin(2x)-sin(6x) dx`
`Q=1/4 (-cos(4x)/4-cos(2x)/2+cos(6x)/6)`

`Q=int (dx)/sqrt(tanx)`
`x=u/2`
`(dx)/(du)=1/2`
`Q=1/2 int (du)/sqrt(tan(u/2))`
`tan(u/2)=t`
`du=(2dt)/(1+t^2)`
`Q=1/2 int (2/(1+t^2))/(sqrt(t)) dt`
`Q=int (dt)/(sqrt(t)(1+t^2))`
`p^2=t`
`(dt)/(dp)=2p`
`Q=int (2p)/(p(1+p^4)) dp`
`Q=2 int (dp)/(p^4+1)`
`hal shode`
`link`
`2(sqrt(2)/8(ln((p^2+sqrt(2)p+1)/(p^2-sqrt(2)p+1))+2arctan(sqrt(2)p+1)-2arctan(1-sqrt(2)p)))`
`p=sqrt(tanx)`

`Q=int xsqrt((x-1)/(x+1)) dx`
`t^2=(x-1)/(x+1)`
`x=(1+t^2)/(1-t^2)`
`(dx)/(dt)=(4t)/(1-t^2)^2`
`Q=(1+t^2)/(1-t^2)t(4t)/(1-t^2)^2 dt`
`Q=int 4(t^4+t^2)/(1-t^2)^3 dt`
`4(t^4+t^2)/(1-t^2)^3=A/(t+1)+B/(t+1)^2+C/(t+1)^3+D/(t-1)+E/(t-1)^2+F/(t-1)^3`
`A=1/2,B=-3/2,C=1,D=-1/2,E=-3/2,F=-1`
`Q=int 1/(2(t+1))-3/(2(t+1)^2)+1/(t+1)^3-1/(2(t-1))-3/(2(t-1)^2)-1/(t-1)^3 dt`
`Q=1/2((6t^3-2t)/(t^2-1)^2+ln((t+1)/(t-1)))`
`|t=sqrt((x-1)/(x+1))`

`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))`