Reduction formula integration method help

tataraperz
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Homework Statement


Well, I've been doing a lot of research about the reduction formula as a method for resolving integration problems. However, not much information on the topic is to be found about the topic. The only things i get to find are examples with the formulas already given, which is cool, but still not what i need.

The Question is ¿is there a general method to find the reduction formula for any given u^n function?
Thanks.
 
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It depends on the function it self.
Personally, I think integration by parts is the famous way of finding the reduction formulas.
 

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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