Integration and radicals (3)

$\begin{aligned} I & = \int_0^1 \frac {x^2}{\color{#3D99F6}\sqrt {x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x \cdots}}}}} dx & \small \color{#3D99F6} \text{See note.} \\ & = \int_0^1 \frac {x^2}{\color{#3D99F6} x^{e-2}} dx \\ & = \int_0^1 x^{4-e} dx \\ & = \left[\frac {x^{5-e}}{5-e} \right]_0^1 \\ & = \frac 1{5-e} \end{aligned}$

$\implies k = \boxed{5}$

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Note: Reference eqns 33-35

Note that:

$\begin{aligned} e & = 1 + \frac 1{1!} + \frac 1{2!} + \frac 1{3!} + \frac 1{4!} + \frac 1{5!} + \cdots \\ & = 1 + 1 + \frac 12 \left(1 + \frac 13 \left(1 + \frac 14 \left(1 + \frac 15 \left(1 + \cdots \right) \right) \right) \right) \\ \implies x^{e-2} & = \sqrt {x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x \cdots}}}} \end{aligned}$