# Integrating 1/xln(x) using integration by parts

by ductruffe
Tags: integration parts ln
 P: 3 1. The problem statement, all variables and given/known data Integrating 1/xlnx by parts... 2. Relevant equations Find the integral of 1/xlnx The question asks to solve by substitution, which I can do and results in ln(ln(x)) + c It then asks to compute using integration by parts, and then to explain how it can be true (because it will compute something different to substitution). 3. The attempt at a solution I = uv - int (v dU) let u= 1/lnx du = 1/x(lnx)^2 let dv = 1/x, v = lnx Sub into the parts formula I = lnx* 1/lnx - int (lnx/x(lnx)^2) I = lnx/lnx - int (1/xlnx) <--- what we started with I = 1 - int (1/xlnx) This is 1 - I, the integral we began with... I've bene shown this trick where you can go.. I = 1- I 2I = 1 I = 1/2 I'm not sure if this is correct, but I would appreciate any help Thank you 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution
 P: 241 It is correct
 P: 3 Thank you for the quick reply. I'm also asked to explain how I = 1/2 can be true, when using substitution yields ln(ln(x)). This is basically where I am stuck. Thank you
P: 241

## Integrating 1/xln(x) using integration by parts

ln(ln(x)) is correct.
Regarding Integration by parts, you have missed -ve sign in differentiating u
It will be
I=1+I
resulting in 1=0 which is wrong.
P: 1,389
 Quote by n.karthick ln(ln(x)) is correct. Regarding Integration by parts, you have missed -ve sign in differentiating u It will be I=1+I resulting in 1=0 which is wrong.
No, it results in 1 = C.
 P: 3 I can see I've missed the negative which changes it quite a bit. Is the integration by parts correct for C=1? It hasn't really solved the integral, or am I missing something here?
 P: 1,389 Typically, you'll see this problem as an example of why it's so important to remember the constant of integration, because otherwise you end up with nonsense like 1 = 0.

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