On inconsistency in Integrate command

Questions
,
1

It seems there exists an inconsistency in the integration algorithms. For example, if you do Integrate[x/(x^2+1),{x,-Infinity, Infinity}], you get 0, but if you compute Integrate[1/Log[x],{x,0,2}], you get divergence.

2

For example, if you do Integrate[x/(x^2+1),{x,-Infinity, Infinity}],
you get 0, but if you compute Integrate[1/Log[x],{x,0,2}], you get
divergence.

That is not what I get:

3497Untitled.png
3497Untitled.png630Ă—428 7.99 KB

3

General rule: indicate version of software.

4

In[4]:= Integrate[x/(x^2 + 1), {x, -b, b}]

Out[4]= 0

which is what I expect due to the symmetry of the integrand under sign change.

I guess that integral goes to infinity - infinity as b goes to infinity.

In[5]:= $Version

Out[5]= "10.0 for Microsoft Windows (64-bit) (December 4, 2014)"
5

Here comes annother gereral rule:

One always should have a rough idea about the function which is integrated.

In the case of 1/Log[x] there is a pole at x=1. So the integral has to be understood as “Cauchy principal value”:

In[10]:= Integrate[1/Log[x], {x, 0, 2}, PrincipalValue -> True]

Out[10]= LogIntegral[2]

In[11]:= % // N

Out[11]= 1.04516

then it works perfect!

Cheers Henrik

In[13]:= $Version

Out[13]= "10.0 for Linux x86 (32-bit) (September 10, 2014)"