Science:Math Exam Resources/Courses/MATH101/April 2009/Question 02

MATH101 April 2009

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Question 02
For what values of p does $\int _{e}^{\infty }{\frac {dx}{x(\ln {x})^{p}}}$ converge?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Try a substitution and then evaluating the integral directly.

Hint 2
Try the substitution $u=\ln(x)$ (Don't forget to substitute the endpoints!)

Hint 3
You might want to isolate the case when $p=1$

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. First, write this integral as $\int _{e}^{\infty }{\frac {dx}{x(\ln {x})^{p}}}=\lim _{b\to \infty }\int _{e}^{b}{\frac {dx}{x(\ln {x})^{p}}}$ Let $u=\ln(x)$. Then $du=dx/x$, $u(e)=\ln(e)=1$ and $u(b)=\ln(b)$. Then ${\begin{aligned}\int _{e}^{\infty }{\frac {dx}{x(\ln {x})^{p}}}&=\lim _{b\to \infty }\int _{e}^{b}{\frac {dx}{x(\ln {x})^{p}}}\\&=\lim _{b\to \infty }\int _{1}^{\ln(b)}{\frac {du}{u^{p}}}\\\end{aligned}}$ At this point, we see that we need to break this up into two cases. The integral above is different when $p=1$ and when $p\neq 1$. So suppose $p\neq 1$. Then we have ${\begin{aligned}\int _{e}^{\infty }{\frac {dx}{x(\ln {x})^{p}}}&=\lim _{b\to \infty }\left.{\frac {u^{-p+1}}{1-p}}\right|_{1}^{\ln(b)}\\&=\lim _{b\to \infty }{\frac {\ln(b)^{-p+1}}{1-p}}-{\frac {1}{1-p}}\end{aligned}}$ This first term will tend to $0$ provided that $1-p<0$ so whenever $p>1$. This term will tend to infinity when $p<1$ since the logarithmic term stays in the numerator. To take care of the $p=1$ case that we have left out, we have ${\begin{aligned}\int _{e}^{\infty }{\frac {dx}{x(\ln {x})^{1}}}&=\lim _{b\to \infty }\int _{e}^{b}{\frac {dx}{x\ln {x}}}\\&=\lim _{b\to \infty }\int _{1}^{\ln(b)}{\frac {du}{u}}\\&=\lim _{b\to \infty }\left.\ln(u)\right|_{1}^{\ln(b)}\\&=\lim _{b\to \infty }\ln \ln(b)-\ln(1)\end{aligned}}$ and this diverges. So our integral converges when $p>1$ and diverges when $p\leq 1$ as required.

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