Expert Answers

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You should solve the given integral using substitution such that:

`ln x = t => 1/x dx = dt`

Changing the variable yields:

`int (dx)/(x*(ln x)^p) = int (dt)/(t^p)`

Using the negative power property yields:

`int (dt)/(t^p) = int (t^(-p))dt = t^(-p+1)/(-p+1) + c`

Substituting back `ln x`  for t yields:

`int_1^2 (dx)/(x*(ln x)^p) = (ln x)^(1-p)/(1-p)|_1^2`

`int_1^2 (dx)/(x*(ln x)^p) = (ln 2)^(1-p)/(1-p) -(ln 1)^(1-p)/(1-p)`

Since `ln 1 = 0`  yields:

`int_1^2 (dx)/(x*(ln x)^p) = (ln 2)^(1-p)/(1-p)`

Hence, evaluating the given definite integral yields `int_1^2 (dx)/(x*(ln x)^p) = (ln 2)^(1-p)/(1-p).`

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