What is the Solution to a Challenging Integration Homework Problem?

MillerL7
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I attached just one question...My tutor and the TA could not figure it out or help me get started on it...can someone help me get started? Thank you so much!
 

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First Please type the problem in rather that posting a "word" file. It's not that difficult and many people won't open a "word" attachement for fear of viruses. I wouldn't if I didn't have very strong virus protection.

As for the problem itself, Draw a picture. While you don't know the exact "form" of f, you know that it passes through the points (5, 13) and (11, 6) and is decreasing so you can sketch a possible graph for f. You also know that the integral given is the area under that curve- the area between that curve and the x-axis. Draw the boundaries of that region.

Now, f-1 just "swaps" x and y so the integral of f-1 you are looking for is the area between that curve and the y- axis. Draw the boundaries and conpare the two areas.
 
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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