Proving Equality Using Lebesgue Monotone Convergence Theorem

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Homework Statement


use the Lebesgue monotone convergence theorem to prove the following equality (attached)


Homework Equations


Lebesgue monotone convergence theorem


The Attempt at a Solution


i tried to identify a suitable monotonic sequence,from power series.
 

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hedipaldi said:

Homework Statement


use the Lebesgue monotone convergence theorem to prove the following equality (attached)


Homework Equations


Lebesgue monotone convergence theorem


The Attempt at a Solution


i tried to identify a suitable monotonic sequence,from power series.

The integral is not clear (poor quality image). Is it
\int_0^1 \frac{x^p}{x-1} \log (x) dx? Do you mean dm(x) = dx?

RGV
 
yes,this is correct.
I already solved .Thank's a lot.
 

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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