`Ifint_(log2)^x(dx)/(sqrt(e^x-1))=pi/6,"then " x " is equalto"` (a)4 (b) log 8 (c) log 4 (d) none of these

Options

A. `log 4`
B. `In 8`
C. `In 4`
D. none of these

Answer

C

Explanation

`(pi)/6=int_(log2)^(x)(dx)/(sqrt(e^(x)-1))=int_(log2)^(x)(e^(x//2)dx)/(e^(x//2)sqrt((e^(x//2))^(2)-1))`
`=2[sec^(-1)e^(x//2)]_(log2)^(x)`
`=2[sec^(-1)e^(x//2)sec^(-1)sqrt(2)]`
or `(pi)/12+(pi)/4=sec^(-1)e^(x//2)`
or `(pi)/3 sec^(-1)e^(x//2)`
or `e^(x//2)=2`
or `x//2=log2`
or `x=log4`