If `int_(log" "2)^(x)(du)/((e^(u)-1)^(1//2))=(pi)/(6)`, then `e^(x)` is equal to

Options

A. 1
B. 2
C. 4
D. `-1`

Answer

C

Explanation

Let `l=int_("log 2")^(x)(e^(u))/(e^(u)(e^(u)-1)^(1//2))du`
Put `e^(u)-1=t^(2)`
`impliese^(u)du=2t" "du`
`thereforel=int_(1)^(sqrt(e^(x)-1))(2t)/((t^(2)+1)t)dt`
`=2int_(1)^(sqrt(e^(x)-1))(dt)/((1+t^(2)))`
`=2[tan^(-1)t]_(1)^(sqrt(e^(x)-1))`
`=2[tan^(-1)sqrt(e^(x)-1)-(pi)/(4)]`
`=(pi)/(6)` [given]
`impliestan^(-1)sqrt(e^(x)-1)=(pi)/(12)+(pi)/(4)=(pi)/(3)`
`implies sqrt(e^(x)-1)=tan((pi)/(3))=sqrt(3)`
`implies e^(x)=3+1=4`.