मान ज्ञात कीजिए : `int(dx)/(x-1sqrt(x^(2)+x+1))`

Answer

`1/sqrt3 log | ((1)/(x-1) +1/2) + sqrt((12(1/(x-1)+1/2)^2 + 1)/(12))|` +c

Explanation

माना,

`I= int (dx)/( (x-1) sqrt( x^ 2 +x+1))`

`x-1= 1/ t` रखने पर

` dx=-1/t^ 2dt`

`I= int (-1/t^ 2 dt) /(1/t sqrt ( (1/t +1)^ 2 +( 1/t +1)+1))`

`=- int dt/ (sqrt (3 t^ 2 +3t+1)`

`=- 1/ sqrt 3 int (dt)/( sqrt ((t+ 1/ 2 )^ 2 +1/12))`

` =- 1 /sqrt 3 log|(t+1/2)+ sqrt ((t+1/2)^ 2 +1/12) | +c`

` =- 1 /sqrt 3 log|( 1 /(x-1) + 1 /2 )+ sqrt( (12( 1 /(x-1) + 1 /2 )^ 2 +1 ))/12 |+c`