When is x^(-1)(C_1+C_2 ln x) in L^2(0,1)?

[LagrangeEuler]

If some function is element of space $L^2(0,1)$ then
$$\int^1_0|f(x)|^2dx< \infty$$. What in the case when it is not so simple to calculate this integral. For example $f(x)=x^{-1}(C_1+C_2 \ln x)$. How to find is it this function in $L^2(0,1)$ for some $C_1,C_2$?

 

[mathman]

It is never (other than both constants = 0). Integral diverges at x=0.

 

[Amal Chacko]

Please help me to define Relation between Lebesgue Differentiation and Lebesgue integration?

 

[mathman]

Please help me to define Relation between Lebesgue Differentiation and Lebesgue integration?

I have never seen the term Lebesgue differentiation. Lebesgue integration is a theory of integration based on measure theory.

 

[WWGD]

Please help me to define Relation between Lebesgue Differentiation and Lebesgue integration?

This is not quite the same as Lebesgue differentiation, bit it is kind of close to it:

https://en.wikipedia.org/wiki/Lebesgue's_density_theorem

 

[Hawkeye18]

Here is the Lebesgue differentiation theorem,
https://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem

 

[WWGD]

Ah, yes, Hawkeye's link is clearly better, more direct than mine.