if `I=int(log(t+sqrt(1+t^2)))/sqrt(1+t^2)dt=1/2(g(t))^2+c` then `g(2)` is (A) `2 log(2+sqrt5)` (B) `log (2 + sqrt 5)` (C) `1/sqrt5 log (2 + sqrt5)` (D) `1/2 log ( 2 + sqrt 5)`

Options

A. `(1)/(sqrt5)log(2+sqrt5)`
B. `(1)/(1)log(2_sqrt5)`
C. `2log(2+sqrt5)`
D. `log(2+sqrt5)`

Answer

D

Explanation

`intlog(t+sqrt(1+t^(2)))/(sqrt(1+t^(2)))dt=(1)/(2)(g(t))^(2)+C" …(i)"`
`"Let "l=intlog(t+sqrt(1+t^(2)))/(sqrt(1+t^(2)))dt`
`"Put "u=log(t+sqrt(1+t^(2)))`
`rArr" "du=(1)/(t+sqrt(1+t^(2)))xx(1+(1)/(2sqrt(1+t^(2)))xx2t)dt`
`rArr" "du=(1)/(t+sqrt(1+t^(2)))xx(sqrt(1+t^(2))+1)/(sqrt(1+t^(2)))dt`
`rArr" "du=(dt)/(sqrt(1+t^(2)))`
`therefore" "l=int udu`
`rArr" "l=(u^(2))/(2)+C`
`rArr" "l=(1)/(2)[log(t+sqrt(1+t^(2)))]^(2)+C" ...(ii)"`
From Eps. (i) and (ii), we get
`g(t)=log(t+sqrt(1+t^(2)))`
`therefore" "g(2)=log(2+sqrt5)`