1.

If `int_(log" "2)^(x)(du)/((e^(u)-1)^(1//2))=(pi)/(6)`, then `e^(x)` is equal toA. 1B. 2C. 4D. `-1`

Answer» Correct Answer - C
Let ` l = int _(log2)^(x) e^(u)/(e^(u)(e^(u)-1)^(1//2) )du `
Put `e^(u) - 1 = t^(2) rArr e^(u) du = 2t dt`
` :. l = int _(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 ]
` rArr tan^(-1) sqrt(e^(x)-1) = pi/12 + pi/4 = pi/3 `
` rArr sqrt(e^(x)-1) = tan (pi/3) = sqrt(3)`
` rArr e^(x) = 3+1 = 4`


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