How can a positive integrand integrate to 0? I integrated (log x)/(1+x^2) from 0 to infinity with residue calculus and got... 0. This also agrees with Wolfram Alpha. How can this be?

st3he1d0t

st3he1d0t

Answered question

2022-09-01

How can a positive integrand integrate to 0?
I integrated log x 1 + x 2 from 0 to infinity with residue calculus and got... 0
This also agrees with Wolfram Alpha.
How can this be?
Is it due to the behavior of log ( x ) near the origin? Like a cancellation effect?
Thanks,

Answer & Explanation

Matthew Benton

Matthew Benton

Beginner2022-09-02Added 9 answers

After the substitution u = 1 x you get
1 log x 1 + x 2 d x = 1 log ( 1 x ) 1 + 1 x 2 ( 1 x 2 ) d x = 1 0 log u 1 + u 2 d u = 0 1 log u 1 + u 2 d u
Yes, the two integrals cancel out.
If you calculate your integral the same way, you get that it is equal to minus itself...

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