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)/(3)tan^(-1)((2sqrtx-1+1)/(sqrt(3)))+C`
D. none of these

Answer

c

Explanation

Let `x-1=t^(2)`. Then,
`int(1)/(x+sqrt(x-1))dx=int(t)/(t^(2)+t+1)dt=int((2t +1)-1)/(t^(2)+t+1)dt`
`=int(2t+1)/(t^(2)+t+1)dt =int(1)/(t+(t+(1)/(2))^(2)+((sqrt(3))/(2))^(2))dt`
`=log (t^(2)+t+1)-(2)/(sqrt(3))tan^(-1)((2t+1)/(sqrt(3)))+C`
`=log (x+sqrt(x-1))-(2)/(sqrt(3))tan^(-1)((2sqrt(x-)+1)/(sqrt(3)))+C`