The value of `int(1)/(x+sqrt(x-1))dx`, is

Options

A. `log(x+sqrt(x-1))+sin^(-1)(sqrt((x-1)/(x)))+C`
B. `log(x+sqrt(x-1))+C`
C. `log(x+sqrt(x-1))-(2)/(sqrt3)tan^(-1)((2sqrt(x-1)+1)/(sqrt3))+C`
D. None of the above

Answer

C

Explanation

Let `l=int(dx)/(x+sqrt(x-1))`
Put`" "x=t^(2)+1 rArr dx=2tdt`
`therefore" "l=int(2t)/(t^(2)+t+1)dt=int(2t+1)/(t^(2)+t+1)dt-int(1)/(t^(2)+t+1)dt`
`=log(t^(2)+t+1)-int(1)/((t+(1)/(2))^(2)+((sqrt3)/(2))^(2))dt`
`=log(t^(2)+t+1)-(2)/(sqrt3)tan^(-1)((2t+1)/(sqrt3))`
`=log(x+sqrt(x-1))-(2)/(sqrt3)tan^(-1)((2sqrt(x-1)+1)/(sqrt3))+C`