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if int(log2->x) (du)/(e^u-1)^(1/2) =pi/6...

if `int_(log2->x) (du)/(e^u-1)^(1/2) =pi/6` then `e^x=`

A

1

B

2

C

4

D

`-1`

Text Solution

Verified by Experts

The correct Answer is:
C
`int_(log2)^(x)(du)/((e^(u)-1)^(1//2))=(pi)/(6)`
`"Put "e^(u)-1=t^(2)`
`rArr" "int_(1)^(sqrt(e^(x)-1))(2t)/(1+t^(2))dt=(pi)/(6)`
`rArr" "2(tan^(-1)t)_(1)^(sqrt(e^(x)-1))=(pi)/(6)`
`rArr" "tan^(-1)sqrt(e^(x)-1)-(pi)/(4)=(pi)/(12)`
`rArr" "sqrt(e^(x)-1)=tan.(pi)/(3)rArr sqrt(e^(x)-1)=sqrt3rArr e^(x)=4`
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