Help with Fourier Integrals for University Assignment

fahd
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hi there.
can neone help with this
i am studying Fourier transforms and Fourier integrals at univ.i came across this question which looks pretty simple..however after doing all that boundary value crap..im totallt confused..pLEASE HELP ME!

QUESTION is attached in the pic...pleasezz help...gotta submit it tomorrow!
 

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fahd, you've been here for about a month and a half, and you know the rules. Post what you've done and we'll help you from there.
 

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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