The value of `int_(1)^(2)(dx)/((x+1)sqrt(x^(2)-1))` is

Options

A. 1
B. `(1)/(sqrt(3))`
C. `(2)/(sqrt(3))`
D. `(-2)/(sqrt(3))`

Answer

B

Explanation

let `l=int_(1)^(2)(dx)/((x+1)sqrt(x^(2)-1))`
Put x=sec`thetaimpliesdx=secthetatanthetad theta`
`thereforel=int_(0)^(pi//3)(sec theta tan theta)/((sec theta+1)tantheta)d theta`
`=int_(0)^(pi//3)(1)/(1+cos theta) d theta`
`=int_(0)^(pi//3)("sec"^(2)(theta)/(2))/(2)d theta`
`=(1)/(2)[("tan"(theta)/(2))/((1)/(2))]_(0)^(pi//3)=(1)/(sqrt(3))`