Simple algebra-calculus question

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1. Homework Statement , and relevant equations

How is A6 arrived at and *how* did they get the "Q" in A6?
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Homework Equations



See above
3. The Attempt at a Solution

Objective: Rid A1 of L. They simply substituted "C" in A2 into A1. But how did they get rid of the L? I don't quite get the algebraic manipulation...please help
 
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What I specifically do not understand is how the exponents, alpha and beta, have been manipulated to be ratios. The only way I think you can do that is by using logarithms, but I do not quite know how.

Anyone have any ideas?
 
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Thread 'Prove that the integral is equal to ##\pi^2/8##'

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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