We want to find the solution for the indefinite integral from:
1. $∫ (\ln x)\ dx$
2. $∫ 1/(\ln x)\ dx$
3. $∫ 1/(x.\ln x)\ dx$
4. $∫ 1/(x^{2}.\ln x)\ dx$
We have to keep in mind that:
$∫ 1/x\ dx = \ln x +\ C$
To solve those, we can use different methods:

Integral by parts
$∫ u.dv = u.v  ∫ v.du$
 Substitution  of $1/u$
$∫ 1/{u}\ du = \ln u +\ C$
 Using (table lookup, or probably remembering some) integral table
1^{st} problem
$∫ (\ln x)\ dx$
2^{nd} problem
$∫ 1/(\ln x)\ dx$This is a special logarithmic integral.
So the solution would be (using integral table):
Or (using jqMath — great with Firefox or other browser which supports
MathML
)
$$∫{{dx}/{\ln x}}={\ln \ln x} + \ln x +Σ↙{k=2}↖∞{(\ln x)^k/{k.k!}}$$
3^{rd} problem
$∫ 1/{x.(\ln x)}\ dx$
Last problem
$∫ 1/{{x^2}.(\ln x)}\ dx$Let's take a look again at the integral table. It magically reveals this form:
Or
You can substitute the:
Note: you don't have to "solve" the second integral, it's an iterating stuff. BUT, if you have so much idle time, you can tinker that.
You can go to WolframAlpha to find out the graphs of those problems: over there
So, in conclusion, finding the right method and "remembering" integral table are important to solve those kind of problems.
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