A logarithm integral Calculate the integral int_(0)^(1)(ln(sqrt(x)-sqrt(1-x)))/(sqrt{x} } dx and show the value is negative.

Kaydence Villegas

Kaydence Villegas

Open question

2022-08-20

A logarithm integral
Calculate the integral
0 1 ln ( x 1 x ) x   d x
and show the value is negative.

Answer & Explanation

Mohammad Hampton

Mohammad Hampton

Beginner2022-08-21Added 7 answers

This integral cannot be real. The issue in Samrat's proof is that log ( x 1 x ) has sort of jump at x = 1 / 2
Let us use the standard branch cut so that log ( x ) = log x + π i for x > 0. Then we can write
0 1 log ( x 1 x ) x d x = 0 1 / 2 log ( 1 x x ) + π i x d x + 1 / 2 1 log ( x 1 x ) x d x = 2 π i + I 1 + I 2 ,
where
I 1 = 0 1 / 2 log ( 1 x x ) x d x and I 2 = 0 1 / 2 log ( 1 x x ) 1 x d x .
For I 1 , integration by parts shows that
I 1 = [ ( 2 x 2 ) log ( 1 x x ) ] 0 1 / 2 + 1 2 0 1 / 2 2 x 2 1 x x ( 1 1 x + 1 x ) d x .
(Here, the bizarre choice for the antiderivative is introduced to cancel out the logarithmic singularity.) Likewise,
Adding them together,
I 1 + I 2 = 0 1 / 2 ( 1 1 x + 1 x ) d x = 0 1 d x x = 2.
Therefore the final answer is
0 1 log ( x 1 x ) x d x = 2 + 2 π i .
Expositur3e

Expositur3e

Beginner2022-08-22Added 1 answers

I = 0 1 ln ( x 1 x ) x d x = 2 x ln ( x 1 x ) | 0 1 0 1 x + 1 x 1 x d x
The first part is 0 and the second integral is
1 1 / 2 0 1 x + 1 x x ( 1 x ) d x = 1 1 / 2 0 π / 2 2 sin θ cos θ sin θ cos θ d θ = 1 π / 2

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