Stuck on the integration. Help me please

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Homework Statement



Hello! I am in the middle of the integration process. Can you give me suggestions on how to simplify the last equation? Thank you!

Please see the attached snapshot of my work.
 

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If I do some algebra, I get

(a+b)^3 - (a-c)^3 = 3a^2(b-c) +3a(b^2-c^2) +b^3-c^3

Does that help?
 
Hi! I did it and it yields results different from the book answer.
Can here be a simpler method of simplification?
 
Kinetica said:

Homework Statement



Hello! I am in the middle of the integration process. Can you give me suggestions on how to simplify the last equation? Thank you!

Please see the attached snapshot of my work.

I just glanced at the problem but I think this really very extremely super-duper basic elementary algebra trick would help:

$$ x^3-y^3=(x-y)(x^2+xy+y^2$$
 

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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