How Do I Apply Binomial Expansion for x^{-1/2}(2-x)^{-1/2} Approximations?

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Homework Statement


<br /> x^{-1/2}(2-x)^{-1/2}<br />

1) approximate to lowest order in x
2) approximate to next order in x

Do I apply the binomial expanion?

Homework Equations


The Attempt at a Solution

 
Last edited:
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That seems like a good idea.
 
<br /> <br /> x^{-1/2}(2+x/2)<br /> <br />

<br /> <br /> 2x^{-1/2}(1/2+x^{1/2})<br /> <br />

lowest order is 2x^(-1/2)?
 

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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