Showing positivity of an integral

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SUMMARY

The discussion centers on demonstrating the positivity of the integral involving a piecewise linear continuous function f and its derivative f', constrained by f' ≤ |2|. The integral of interest is ∫[f'(s)H(s)]ds outside the interval [-1, 1], where H(s) is defined as 0 between -1 and 1 and as s[ln(s) ± √(s² - 1)] ± √(s² - 1) otherwise. The user seeks to establish that this integral is positive and considers using the relationship ∫[f(s)[ln(s) ± √(s² - 1)]] for simplification. The discussion highlights the utility of WolframAlpha for integral calculations.

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hnh
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Hello, I have a tricky integral to show positivity of. Here are the knowns: f is piecewise linear continuous, f' <= abs{2}, H(s) = 0

between -1 and 1, and s[ln(s)+- sqrt{s^2 -1}] +- sqrt{s^2 -1} otherwise.

I wish to show that int [f'(s)H(s)]ds outside interval [-1,1] is positive. One suggestion is to show that the integral
is = int [f(s) [ln(s) +- sqrt{s^2-1}]] but I am unable to do this.

Any discussion is appreciated. I am an algebra grad student. This is needed for evaluation of other integrals and I am stuck. Thank you
 
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