Short improper integral question (how to rewrite?)

vantz
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Homework Statement


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Homework Equations


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The Attempt at a Solution


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I'm trying to rewrite the integral as shown

Most probably a real simple answer

Thank you
 
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In the first integral, try substituting x for -x.
 
I don't understand, wouldn't that be illegal to do?
 
vantz said:
I don't understand, wouldn't that be illegal to do?

Yes, it would. But Compuchip meant to do a u-substitution. u=(-x), du=(-dx).
 
Exactly. And don't forget the integration boundaries as well.
 
I have it now. Thank you
 

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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