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Problem 58 Hard Difficulty

Find the values of $ p $ for which the integral converges and evaluate the integral for those values of $ p $.

$ \displaystyle \int_e^\infty \frac{1}{x (\ln x)^p}\ dx $

Answer

converges to $\frac{1}{p-1}$ only if $p > 1$

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Video Transcript

Hello. Welcome to this lesson. And this lesson. You're looking for the value of B for which the whole integral converges. So list left, learn x it cost to you So that do you will be caught one of our x dx. And this implies that X do you is a call to the X. So if if Lennox go to you, then we have you. It costs too. 11 XC court e and also you it cost to infinity one x sick or to infinity So means that the whole of the limits we change from e 21 them phones if they will, may not infinity. So that is one over. The whole of the DX is replaced by xD. You so you have excellent place off Len X. Who put you there? Yeah. Okay. So, like, this can cross out that. Wow. Okay, right. So wow. Yeah. Uh huh. By integration. Oh, to infinity. New negative B. Oh, right. Do you village point? So let is equal to negative p last one. Yeah. Wow. Yes. To sell a limit. So a limit than you. Mhm. Okay. Okay. Yeah, And that is from one to t. No So the fourth thing Now we comes. Let's put one in the one and t in there. So the limit, he approaches infinity. Oh, see? Okay. Yeah, of T negative people. Last one book. Okay, then. Minus what? One here. Then this becomes negative people. Last one. Okay, so there is a limit T purchase Infinity. Yeah. So the negative comes down so that I can have p minus one looking in there? Yeah. So here, uh, if he is greater than one, if p is better than one, okay, It means that the whole thing here would be a denominator to. And if a denominator is better than zero means that it approaches zero, a denominator is greater than zero. The whole of their, the whole of the fraction approaches there. Okay, so here we can have zero the whole of this with up put zero. Okay, then this is both one of, uh, t minus one king when p is greater than one. All right, so if P is positive, then it would give room for the whole of this to approach there. All right, so this is the value of the p that makes the whole thing converge. Okay, so thanks your time. This is the end of the