Solving Troublesome Project Question - Help Needed

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I have a project to do (actually, it's almost complete). I'm just having trouble on one question. I'm not sure if it is as hard as it appears, or if it is just poorly written. Anyway the question I have is http://www.students.niu.edu/~z086659/question.pdf" .

If you have any ideas or hints, please respond.

Thank you.
 
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I'm afraid that doesn't make a whole lot of sense to me.

We are given a table which we are told shows, for each population P, the birth and death rates, B(p) and \delta(P). The value in the table for B(P) is, as is correct for a rate, between 0 and 1, but \delta(P) is not- it is a number in the thousands. Further, even though we are given \delta(P) in the table, part of the question says we are to "find \delta(P)".
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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