Problem with a taylor serie expansion

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Homework Statement
Look at the image
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Tayolr development
Greetings
https://www.physicsforums.com/attachments/295843

I really don´t agree with the solution
https://www.physicsforums.com/attachments/295846

as I calculated fxy I got
fxy=xyexy
f(0,1)=0 so x(y-1) should not appear in the solution
am I wrong?

thank you!
 
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thank you I solved the problem it was my wrong calculation!
 

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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