Re: Integral Inequality

Grigorios Kostakos <grigkost@xxxxxxxxxxxxx> wrote:
>Any ideas how to prove that
>\int_(-x)^(x)_(ln(exp(x-t)+1))_dt < exp(x)+3/2 ? For every real x>0.

Changing variable in the integral to exp(x-t) and setting y = exp(2x),
the statement is equivalent to proving that

sqrt(y) + 3/2 - int_0^y ln(1+u)/u du

is always > 0 for y > 1.

The derivative of this expression is 1/(2 sqrt y) - ln(1+y)/y,
which is < 0 for y = 1 and has one zero, at 75.03859646576. So it's
only necessary to check that the value is > 0 at this one minimum.


Mike Guy
.

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