Evaluating and integral using random numbers

ToxicBug
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Homework Statement


Hello! I need to evaluate the following integral by using random numbers. I just need help with rewriting the boundaries of the integral to both be [0,1]


Homework Equations


<br /> \iint \frac{\,x_1\,x_2}{\,(1+2x_1)^2\exp(x_2^2)}\,dx_1\,dx_2<br />
The integral on the following region:
<br /> 2&lt;x_1&lt;x_2&lt;\infty<br />


The Attempt at a Solution


I've no idea how to transform the double integral, but I know that for a single integral I need to use the substitution y=e^{-x}
 
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well... I'm not really sure so someone should confirm this, but you can write the intergrand as a product f(x1)g(x2) and then you can just put the f(x1) in front of the one integral, and the g(x2) in front of the other. So now you have the product of two single integrals, can you solve those?
 
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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