T-Cyclic Subspace Generated by Z Using T(f) = f' + 2f in P1(\Re)

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Homework Statement


Find the T-Cyclic subspace generated by Z. V = P1(\Re) T(f) = f' +2f and Z = 2x

Homework Equations


The Attempt at a Solution


so T(1,0) = 2
and T(0,1) = 1 + 2x

so [T]_{}\beta =
( 2 1
0 2 )

So T-cyclic subspace generated by 2x = { 2x, 2 + 4x } ?
 
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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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