In the integral , use the power-series

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Homework Statement



I wrote it on Word for increased clarity.

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Homework Equations



This is in the chapter where we learned about integrals of the "second" type, ones that have a problem as x approaches the upper or lower limit.

The Attempt at a Solution



I know that the power series expansion for log(1+x) is 1 - x2/2 + x3/3 - x4/4 + ... .

But raising that to the q power ... How is that done?
 
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Correct that to log(1+x)=x-x^2/2+x^3/3+... And the only problem is at the lower limit, right? So you can take x to be very close to zero. I'd just say log(1+x)~x. Raising that to the q power should be easy.
 
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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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