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

## Homework Statement

Integrating 1/xlnx by parts...

## Homework 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).

## 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

## The Attempt at a Solution

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It is correct

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

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.

jgens
Gold Member
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.

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?

jgens
Gold Member
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.