Mastering Laplace Transforms: Solving Algebraic Equations with Ease

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I put this in this forum because its actually a laplace transform question

basically, I just can't see how he is factoring out (kp)/(kp-1)
any ideas?
http://img4.imageshack.us/img4/1462/38846579.jpg
 
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It looks to me like a partial fractions decomposition, rewriting
\frac{k_p}{s(s - 1 + k_p)}~=~k_p\left(\frac{A}{s} + \frac{B}{s - 1 + k_p}\right)

The idea is to solve for A and B so that this equation is an identity.
 

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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