Proof of a partial fraction theorem

Kinsama
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I've been looking all over the net since yesterday for a PROOF of a theorem called the "heaviside method" it involves partial fractions and itegration of such. If anyone could show me a proof of it or send me a link of it I would gladly appreciate it, thanks!
 
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Of what, specifically, do you want a proof?

P.S. if you have an algorithm that does something, that is a proof that you can do that something.
 
i want a proof of a theorem called the Heaviside Method, my book tells how to do it, but i need a proof of it for class and its almost impossible to find.
 
Well, if you need it for class, shouldn't you prove it yourself? :-p And I don't even know the theorem you're trying to prove -- I can find references to the algorithm called "Heaviside's method", but you can't prove an algorithm. (The very idea is nonsensical)

My best guess is that you want a proof that the algorithm works as advertised, and I've already given you a hint: just prove that the output of the algorithm is what it claims to be
 
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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