How to Prove the Integral Property for Definite Integrals

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Homework Statement


Today i had a test on definite integrals which i failed. The test paper was given to us so we can practise at home and prepare better for the next one. This is the first problem which i need your help in solving::
Test.JPG


Homework Equations


3. The Attempt at a Solution [/B]
As no points were given for a solution of the below integrals without the proof of the integral property above i need to do that first. I had no idea how to start the proof. I figured i need to use some sort of substitution but i fail to see which and why. Could you give me a hint on how to do this? I know i haven't provided any work done by myself but i can't since i can't start. I didn't have a clue calculus was going to be this hard :/.
Thanks
 
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The key here is that ##sin(x)## is symmetric around ##pi/2##. Hence the substitution ##t=pi/2+x## may be of use. You then see that a term in your new expression should disappear.
 
Incand said:
The key here is that ##sin(x)## is symmetric around ##pi/2##. Hence the substitution ##t=pi/2+x## may be of use. You then see that a term in your new expression should disappear.

I think you probably meant something more like ##x=\pi-t##.
 
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The substitution ##x = \pi - t ## does indeed do the trick for proving the proposition before solving the given integral.
 
nuuskur said:
The substitution ##x = \pi - t ## does indeed do the trick for proving the proposition before solving the given integral.
Yeah i did it with the substitution you proposed but how did you arrive at it?
 
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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