How to perform this integration?

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Homework Statement



The integration:
 

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...I can't view the picture...try re-upping it as a jpeg or something so
 
I re-upload the pic. The integration is that ∫cos(kx)exp[(-αt)K^2] dkfrom zero to infinity
 

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Convert cos(kx) to (exp(ikx)+exp(-ikx))/2. Combine the exponentials and complete the squares in the exponents. Now do a change of variable and use the known result for exp(-Cx^2).
 

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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