Can a Simple Integral Solve 1/(K + x^2)^(3/2)?

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A simple integral?!?

Homework Statement



Integrate 1/(K+ x^2)^(3/2) dx

Homework Equations





The Attempt at a Solution



substitution of some kind - if it had an x on top it would be fine...
 
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Set x=\sqrt{K}Sinh(u) and proceed.
 
arildno loves that hyperbolic substitution!

Myself, I would have immediately thought that "tan^2(\theta)+ 1= sec^2(\theta)" and used the substitution x= \sqrt{K}tan(\theta). I suspect it is just a matter of taste.
 

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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