Determine which values of p the following integrals converge. Give your answer in each case by entering an appropriate inequalitysign (e.g., > or <=) in the box immediately after p and a...

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= ? , ?

Asked on by blasto63

1 Answer | Add Yours

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

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).`

We’ve answered 318,911 questions. We can answer yours, too.

Ask a question