Determine which values of p the following integrals converge. Give your answer in each case by entering an appropriate inequality sign (e.g., > or <=) in the box immediately after p and a numerical value in the second box, to define a range of p values for which the integral converges. If the integral never converges, enter nonefor the numerical value. integrate from 1 to 2 of ((dx)/(x(ln(x))^p)) p= ? , ?

Expert Answers

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You need to evaluate the integral using the following substitution such that:

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

`int (1/x)/((ln x)^p)dx = int (dt)/(t^p)`

Using the negative power property yields:

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

Substituting back `ln x`  for `t`  yields:

`int (1/x)/((ln x)^p)dx = ((ln x)^(1-p))/(1-p) + c`

You need to evaluate the definite integral using the fundamental theorem of calculus such that:

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

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

Since`ln 1 = 0 => ((ln 1)^(1-p))/(1-p) = 0`

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

Hence, since the given integral is not an improper integral, but a definite integral, with finite limits of integration, you only can evaluate it such that `int_1^2 (1/x)/((ln x)^p)dx = ((ln 2)^(1-p))/(1-p).`

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