Linear Algebra Help Projection

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Stumped on #15. I feel like its much easier than I am making it out to be and that maybe I am just over thinking. Any help or leads would be appreciated. Thanks for your time.

http://gem.jsu.edu/service/home/~/MS352%20Test%231%20Take%20Home%20Portion%20June%202012.pdf?auth=co&loc=en_US&id=47121&part=2
 
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I'm getting an error with the link.

Also, if you want some help you have to show what you have tried already.
 
Sorry about the link. And thanks for your time. I have had trouble where to start but I think that I was just reading it wrong. I did #14 just fine. I'm pretty sure that I can figure out #15. #16 I'm not so sure. Any leads would be appreciated. Again thanks for your time.
 

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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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