Definite Integral with u/du subsitition

Knight226
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Homework Statement


Evaluate the indefinite integral by the method shown in Example 5. (the example in the book is just using the u and du substitution)

MathHelp1-1.jpg


Homework Equations


None.

The Attempt at a Solution



The question is right below the instruction and after that question is my steps. Please correct me if I have made mistake somewhere along the way or if I am doing it incorrectly. I get stuck at the last step because I am not sure what I should do to the 2 in front of u^1/2. Please help. Thank you very much.

I don't know how to use the math codes in the forum... so I am attaching a screen shot instead. Thank you.

MathHelp1.jpg
 
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The substitution went just fine. I'm not sure what you want to do with the '2'. Just leave it there and try to integrate both terms du. Can you do that?
 
Thank you.
I apologize for my careless. I got confused with the book example >.<.
 
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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