Understanding Integration: A Simple Explanation

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If one could just explain how the attached answer below was derived I would be greatful. Thanks!
 

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By being very careful with your minus signs! What did you try? Remember that 'a' is a constant and you are integrating with respect to 'x'.
 
The anti-derivative function is a^{2}x - \frac{1}{3} x^{3} . You then evaluate it at x = a and x = -a (or at x = a and double the result, since the integrand function is symmetrical about the y-axis [the anti-derivative is zero at x = 0 ] ).
 

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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