Integrating a Tricky Differential Equation with a Square Root Fraction

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Homework Statement


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Solve the differential equation

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Homework Equations

The Attempt at a Solution


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I just can't integrate that (1+y)^(1/2)/(1+y^2)dy at the end... the other two integrals are trivial.
 

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Try the substitution ##u=\sqrt{1+y}## and see where that gets you.
 
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I actually tried before posting. Once you make the substitution there are two ways to attempt the problem, one by doing u^2 = 1+y and then differentiating, and the other got me to 2*(1+y)^(1/2)du = dy. None of these seemed to make the integral any easier.
 
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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